# Patent

## Title
Data receiver with improved response to long and short term fading

## Abstract

The invention relates to apparatus for the receipt of digital data which is transmitted as a data signal from a remote location and a method by which the receiving apparatus can be designed with a configuration to improve the ability to receive the data signal. The design of the configuration is generated with reference to at least one decision metric which is adapted to suit the known characteristics of the data signal which is being received.

## Background
CROSS-REFERENCE TO RELATED APPLICATIONS
This application claims priority to United Kingdom Patent Application No. 1004946.8 filed Mar. 24, 2010 which is incorporated herein by reference.
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT
Not Applicable
THE NAMES OF THE PARTIES TO A JOINT RESEARCH AGREEMENT
Not Applicable
INCORPORATION-BY-REFERENCE OF MATERIAL SUBMITTED ON A COMPACT DISC
Not Applicable
BACKGROUND OF THE INVENTION
1. Field of the Invention
The invention to which this application relates is a broadcast data receiver apparatus which is provided in a form so as to have an improved response to long and short term fading of the data channels.
2. Prior Art
The demand for increased capacity in data receiving apparatus, and/or reduced information errors continues to grow and there is also an increasing demand for generally efficient operation of wireless communication systems. However, a fundamental constraint to the capacity of the data transmission and receiving system is one or both of the digital receiver and the propagation medium (i.e the channel on which the data is transmitted).
Broadcast signals which are transmitted are typically distorted by atmospheric and environmental impairments, thereby degrading the quality of the signal which is subsequently received. In general, one form of deterioration is fading which is typically caused by physical phenomena such as reflection, diffraction and scattering. According to the number of the scattering phenomena which may be present in a wireless channel, the effect of these phenomena is either rich or poor. This is the case in the occurrence of multipath and shadowing fading, respectively.
It is known from the theory of telecommunications that multipath fading and shadowing of propagated signals, (also known as short-term fading and long-term fading respectively), can be effectively described and represented by statistical distributions.
Conventionally used distributions such as those known as Rayleigh, Weibull and Nakagami -m, -n, -q distributions, are adequate when describing multipath fading. However, they are found not to be optimal and/or robust because they do not additionally consider the simultaneous occurrence of the shadowing effect.
Another known distribution known as the Suzuki statistical distribution has the advantage that it is a composite Rayleigh/Lognormal distribution. As a consequence, it is capable of describing at once both the multipath and the shadowing of communication signals which occur simultaneously during wireless propagation. However, the probability density function (PDF) of the Suzuki distribution is described by the integral of the synthesis of Lognormal and Rayleigh, distribution. An explicit solution to this integral is not known in the prior art and thus it has not been possible to implement a receiver for a digital data system which is operable over such a fading channel.
An existing method of compensating for shadowing is by providing the receiver with empirical data, i.e. actual data, relating to the geographical area in which the receiver is to be used. However, it, is time-consuming to produce such data, the data is not always specific enough to the receiver location, and if the receiver is moved to a new location, new empirical data has to be obtained.
An aim of the present invention is to provide a receiver which can characterise accurately the signal degradation caused by the presence of multipath fading and shadowing. Another aim is to provide a method whereby the characterisation of the signal degradation can be accurately determined.
BRIEF SUMMARY OF THE INVENTION
In one aspect of the invention, there is provided a broadcast receiver wherein the configuration of the apparatus and/or processing means used to receive the data signal is assessed with respect to a decision metric derived upon utilising the probability density function of a Suzuki distribution in a closed form:




  
    
      p
      
        a
        l
      
    
    ⁡
    
      (
      
        a
        l
      
      )
    
  
  =
  
    
      
        (
        
          5
          2
        
        )
      
      
        1
        
          8
          ⁢
          
            σ
            2
          
        
      
    
    ⁢
    
      
        
          a
          l
        
        ⁢
        
          e
          
            -
            
              
                μ
                2
              
              
                2
                ⁢
                
                  σ
                  2
                
              
            
          
        
      
      
        σ
        ⁢
        
          
            2
            ⁢
            π
          
        
      
    
    ⁢
    
      Γ
      ⁡
      
        (
        
          1
          -
          
            1
            
              8
              ⁢
              
                σ
                2
              
            
          
          -
          
            μ
            
              2
              ⁢
              
                σ
                2
              
            
          
        
        )
      
    
    ⁢
    
      
        (
        
          
            -
            
              1
              
                8
                ⁢
                
                  σ
                  2
                
              
            
          
          -
          
            
              a
              l
              2
            
            2
          
        
        )
      
      
        
          -
          1
        
        +
        
          1
          
            8
            ⁢
            
              σ
              2
            
          
        
        +
        
          μ
          
            2
            ⁢
            
              σ
              2
            
          
        
      
    
  




where σ and μ denote the variance and mean value respectively of the Lognormal distribution, in order to take into account potential impairment of the said data signal.

Thus the receiver can be optimised in a number of conditions such that the requirement for empirically modelling the channel before deploying a new network is removed. As such the receiver has improved performance over fading channels and is adaptive to the location of the receiver.
In one embodiment, where amplitudes are independent with first order probability density functions {pαl(αl)}l=1Lp, and the phase and delay values are known the decision metric to be used which is derived from the Suzuki distribution is:

Λk=ln [p({rl(t)}l=1Lp|sk(t),{θl}l=1Lp,{τl}l=1Lp)]

The decision metric below allows implementation of a receiver design in the case of known phases but unknown delays and amplitudes.

Λk=ln [p({rl(t)}l=1Lp|sk(t),{θl}l=1Lp)]=

From the theory of digital communications, the decision metric for the case of known delays and amplitudes but unknown phases is:




  
    
      
        
          Λ
          k
        
        =
        
          ln
          ⁡
          
            [
            
              p
              ⁡
              
                (
                
                  
                    
                      
                        {
                        
                          
                            r
                            l
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        }
                      
                      
                        l
                        =
                        1
                      
                      Lp
                    
                    ❘
                    
                      
                        s
                        k
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  ,
                  
                    
                      {
                      
                        α
                        1
                      
                      }
                    
                    
                      l
                      =
                      1
                    
                    Lp
                  
                  ,
                  
                    
                      {
                      
                        τ
                        l
                      
                      }
                    
                    
                      l
                      =
                      1
                    
                    Lp
                  
                
                )
              
            
            ]
          
        
      
    
  
  
    
      
        =
        
          κ
          ⁢
          
            
              ∑
              
                i
                =
                1
              
              
                L
                p
              
            
            ⁢
            
              ln
              [
              
                
                  e
                  
                    -
                    
                      
                        
                          a
                          l
                          2
                        
                        ⁢
                        
                          E
                          k
                        
                      
                      
                        N
                        l
                      
                    
                  
                
                ⁢
                
                  
                    I
                    o
                  
                  ⁡
                  
                    (
                    
                      
                        
                          a
                          l
                        
                        
                          N
                          l
                        
                      
                      ⁢
                      
                        
                        
                          
                            y
                            kl
                          
                          ⁡
                          
                            (
                            
                              τ
                              l
                            
                            )
                          
                        
                        
                      
                    
                    )
                  
                
              
              ]
            
          
        
      
    
  




In one embodiment the decision metric derived from the case of known delays but unknown amplitudes and phases is:

Λk=ln [p({rl(t)}l=1Lp|sk(t),{τl}l=1Lp)]

This decision metric allows implementation of a receiver configuration design in the case of known delays but unknown phases and amplitudes.
In one embodiment the decision metric derived from the case of unknown delays, amplitudes and phases is:

Λk=ln [p({rl(t)}l=1Lp|sk(t)]=

This decision metric allows implementation of a receiver configuration design for the case of unknown delays, phases and amplitudes.
In one embodiment the design or configuration of the receiving apparatus include details of coherent detection as well as non-coherent detection.
In one embodiment the designs include details of the possible modulation schemes, typically M-PSK, M-QAM, ASK/PAM, FSK, MSK and OFDM.
In one embodiment the designs include a plurality of shaping pulses, typically square, sinusoidal and cosine.
In one embodiment the broadcast receiver characterises signal impairment optimally by utilising any, or any combination of, the decision metrics described herein.
Typically, according to the level of channel knowledge, the decision metric allows optimisation of the signal reception parameters or design.
In a further aspect of the invention there is provided a method of configuring receiving and/or processing apparatus for at least one data signal received from a remote transmitting location, said method including the steps of:
identifying the geographical location at which the said apparatus is to be used;
identifying the data signal which is to be received and wherein the said apparatus is further configured with reference to a decision metric derived upon utilising the probability density function of a Suzuki distribution in a closed form:




  
    
      p
      
        a
        l
      
    
    ⁡
    
      (
      
        a
        l
      
      )
    
  
  =
  
    
      
        (
        
          5
          2
        
        )
      
      
        1
        
          8
          ⁢
          
            σ
            2
          
        
      
    
    ⁢
    
      
        
          a
          l
        
        ⁢
        
          e
          
            -
            
              
                μ
                2
              
              
                2
                ⁢
                
                  σ
                  2
                
              
            
          
        
      
      
        σ
        ⁢
        
          
            2
            ⁢
            π
          
        
      
    
    ⁢
    
      Γ
      ⁡
      
        (
        
          1
          -
          
            1
            
              8
              ⁢
              
                σ
                2
              
            
          
          -
          
            μ
            
              2
              ⁢
              
                σ
                2
              
            
          
        
        )
      
    
    ⁢
    
      
        (
        
          
            -
            
              1
              
                8
                ⁢
                
                  σ
                  2
                
              
            
          
          -
          
            
              a
              i
              2
            
            2
          
        
        )
      
      
        
          -
          1
        
        +
        
          1
          
            8
            ⁢
            
              σ
              2
            
          
        
        +
        
          μ
          
            2
            ⁢
            
              σ
              2
            
          
        
      
    
  




where σ and μ denote the variance and mean value respectively of the Lognormal distribution, in order to take into account potential impairment of the said data signal.




BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING(S)
Further details of specific embodiments are now described wherein:—
FIG. 1 illustrates a receiver apparatus configuration design for the case of known delay and phase values but unknown amplitude values for a data signal to be received.
FIG. 2 illustrates a receiver apparatus configuration design for the case of known phase value but unknown delay and amplitude values for a received data signal.
FIG. 3 illustrates a receiver apparatus configuration design for the case of a known delay value but unknown phase and amplitude values of a received data signal.
FIG. 4 illustrates a receiver apparatus configuration design for receipt of a data signal for which the delay, phase and amplitude values are unknown.



DETAILED DESCRIPTION OF THE INVENTION
In the transmission of a data signal to one or more receiver apparatus then during a symbol period of ‘Ts’ seconds of the data signal the transmitter transmits a real bandpass signal which can be represented as:

Sk(t)=Re{{tilde over (s)}k(t)}=Re{{tilde over (S)}k(t)ej2nfct}  (1)

where {tilde over (s)}k is the kth complex bandpass signal and {tilde over (S)}k(t) is the corresponding kth, complex baseband signal chosen from the set of M equi-probable message waveforms which represents the transmitted information.

The above signal (1) is transmitted over the fading channel which is characterized by Lp, independent diversity paths. Each of these paths is a time varying channel that attenuates, delays, phase-shifts the signal while it also adds an AWGN source. Thus, the received data signal at the receiving apparatus is in fact a set of “noisy” replicas of the transmitted signal.
It should be noted here that a so-called “one-shot” approach for the transmission of the data signal is assumed to be used herein i.e. a single transmission wherein ISI that would be produced by the presence of the path delays on continuous transmission is ignored. Hence, the received signal will be expressed as follows:

rl(t)=Re{αl{tilde over (s)}k(t−τl)ejθl+nl(t)}=Re{αl{tilde over (S)}k(t−τl)ej(2nfct+θl)+Nl(t)ej2nfct}  (2)

rl(t)=Re{{tilde over (r)}l(t)}=Re{{tilde over (R)}l(t)ej2nfct}l=1,2,3, . . . ,Lp  (3)

where {Nl(t)}hd l=1Lp is a set of statistically independent complex Additive White Gaussian Noise (AWGN) processes, each with PSD 2Nl Watts/Hertz.

The sets {al}l=1Lp, {θl}l=1Lp and {τl}l=1Lp are the random channel amplitudes, phases and delays respectively and they are assumed to be constant over one or two symbol periods i.e. Tg and 2Tg for non-coherent and coherent detection respectively.
Another important assumption is that τ1<τ2< . . . <τLp. The optimum receiver computes the set of a posteriori probabilities p(sk(t)|{rl(t)}l=1Lp), k=1, 2, . . . , M and chooses as its decision that message whose signal sk(t) corresponds to the largest of these probabilities. In addition, the receiver is assumed to be time—synchronised to the transmitted signal.
Since the messages (signals) are assumed to be equi-probable, then, by Bayes' rule, the equivalent decision rule is to choose sk(t) corresponding to the largest of the conditional probabilities (likelihoods)

p({rl(t)}l=1Lp|sk(t)),k=1,2; . . . ,M 

which is the Maximum-Likelihood (ML) decision rule. By making use of the law of conditional probability, each of these conditional probabilities can be expressed by equation (4) as follows:

∫∫∫p({rl(t)}l=1Lp|sk(t),{αl}l=1Lp,{θl}l=1Lp,{τl}l=1Lp)p({αl}l=1Lp,{θl}l=1Lp,{τl}l=1Lp)d{αl}l=1Lpd{θl}l=1Lpd{τl}l=1Lp 

which is dependent upon the degree of knowledge i.e. amount of Channel State Information (CSI) available on the parameter sets) {αl}l=1Lp,{θl}l=1Lp,{τl}l=1Lp.

In case that any of the three parameter sets are assumed to be known, then the statistical averages on that set of parameters need not be performed. Of course, in the case where all parameters are assumed to be known to the receiver, none of the statistical averages in (4) need be performed and as a consequence the ML decision rule simplifies to choosing the largest of:

p({rl(t)}l=1Lp|sk(t),{αl}l=1Lp,{θl}l=1Lp,{τl}l=1Lp),k=1,2, . . . ,M 

Receivers that make use of CSI have been termed self-adaptive in that the estimates of the system parameters are utilized to adjust the decision structure, thereby, improving the system performance by adaptation to variation of the channel changes.
Example 1
Known Amplitudes, Phases and Delays; Non-Coherent Detection
Conditioned on perfect knowledge of the amplitudes, phases and delays, the conditional probability p({rl(t)}l=1Lp|sk(t),{αl}l=1Lp,{θl}l=1Lp,{τl}l=1Lp) is a joint Gaussian Probability Density Function (PDF) that because of the independence assumption on the additive noise components can be written as:




  
    
      
        
          p
          ⁡
          
            (
            
              
                
                  
                    {
                    
                      
                        r
                        l
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    }
                  
                  
                    l
                    =
                    1
                  
                  Lp
                
                ❘
                
                  
                    s
                    k
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
              ,
              
                
                  {
                  
                    α
                    l
                  
                  }
                
                
                  l
                  =
                  1
                
                Lp
              
              ,
              
                
                  {
                  
                    θ
                    l
                  
                  }
                
                
                  l
                  =
                  1
                
                Lp
              
              ,
              
                
                  {
                  
                    τ
                    l
                  
                  }
                
                
                  l
                  =
                  1
                
                Lp
              
            
            )
          
        
        =
        
          
            
              ∏
              
                l
                =
                1
              
              
                L
                p
              
            
            ⁢
            
              
                K
                1
              
              ⁢
              
                e
                
                  -
                  
                    1
                    
                      2
                      ⁢
                      
                        N
                        i
                      
                    
                  
                
              
              ⁢
              
                
                  ∫
                  
                    τ
                    l
                  
                  
                    Ts
                    +
                    
                      τ
                      l
                    
                  
                
                ⁢
                
                  
                    
                      
                      
                        
                          
                            r
                            l
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        -
                        
                          
                            α
                            l
                          
                          ⁢
                          
                            
                              
                                s
                                _
                              
                              k
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  τ
                                  l
                                
                              
                              )
                            
                          
                          ⁢
                          
                            e
                            
                              jθ
                              l
                            
                          
                        
                      
                      
                    
                    2
                  
                  ⁢
                  
                    d
                    t
                  
                
              
            
          
          =
          
            
              ∏
              
                l
                =
                1
              
              
                L
                p
              
            
            ⁢
            
              
                K
                1
              
              ⁢
              
                e
                
                  
                    -
                    
                      1
                      
                        2
                        ⁢
                        
                          N
                          i
                        
                      
                    
                  
                  ⁢
                  
                    
                      ∫
                      
                        τ
                        l
                      
                      
                        Ts
                        +
                        
                          τ
                          l
                        
                      
                    
                    ⁢
                    
                      
                        
                          
                          
                            
                              
                                
                                  R
                                  ~
                                
                                l
                              
                              ⁡
                              
                                (
                                t
                                )
                              
                            
                            -
                            
                              
                                α
                                l
                              
                              ⁢
                              
                                
                                  
                                    S
                                    ~
                                  
                                  k
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    -
                                    
                                      τ
                                      l
                                    
                                  
                                  )
                                
                              
                              ⁢
                              
                                e
                                
                                  jθ
                                  l
                                
                              
                            
                          
                          
                        
                        2
                      
                      ⁢
                      
                        d
                        t
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        5
        )
      
    
  




where K is an integration constant. After simplification of equation (5) we obtain:





  
    
      
        
          
            p
            ⁡
            
              (
              
                
                  
                    
                      [
                      
                        
                          r
                          l
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ]
                    
                    
                      l
                      =
                      1
                    
                    Lp
                  
                  ❘
                  
                    
                      s
                      k
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                ,
                
                  
                    
                      {
                      
                        α
                        l
                      
                      }
                    
                    
                      l
                      =
                      1
                    
                    Lp
                  
                  ⁢
                  
                    
                      {
                      
                        θ
                        l
                      
                      }
                    
                    
                      l
                      =
                      1
                    
                    Lp
                  
                
                ,
                
                  
                    {
                    
                      τ
                      l
                    
                    }
                  
                  
                    l
                    =
                    1
                  
                  Lp
                
              
              )
            
          
          =
          
            
              K
              ⁢
              
                
                  ∏
                  
                    l
                    =
                    1
                  
                  
                    L
                    p
                  
                
                ⁢
                
                  e
                  
                    
                      Re
                      ⁡
                      
                        [
                        
                          
                            
                              α
                              l
                            
                            
                              N
                              l
                            
                          
                          ⁢
                          
                            e
                            
                              -
                              
                                jθ
                                l
                              
                            
                          
                          ⁢
                          
                            
                              y
                              kl
                            
                            ⁡
                            
                              (
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                              )
                            
                          
                        
                        ]
                      
                    
                    -
                    
                      
                        
                          α
                          l
                          2
                        
                        ⁢
                        
                          E
                          k
                        
                      
                      
                        N
                        i
                      
                    
                  
                
              
            
            =
            
              K
              ⁢
              
                
              
              ⁢
              
                e
                
                  
                    
                      ∑
                      
                        l
                        =
                        1
                      
                      
                        L
                        p
                      
                    
                    ⁢
                    
                      Re
                      ⁡
                      
                        [
                        
                          
                            
                              α
                              l
                            
                            
                              N
                              l
                            
                          
                          ⁢
                          
                            e
                            
                              -
                              
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                                l
                              
                            
                          
                          ⁢
                          
                            
                              y
                              kl
                            
                            ⁡
                            
                              (
                              τ
                              )
                            
                          
                        
                        ]
                      
                    
                  
                  -
                  
                    
                      
                        α
                        l
                        2
                      
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                        k
                      
                    
                    
                      N
                      i
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          where
          ,
        
      
    
    
      
        (
        6
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              y
              kl
            
            ⁡
            
              (
              τ
              )
            
          
          =
          
            
              
                ∫
                
                  τ
                  l
                
                
                  
                    T
                    s
                  
                  +
                  
                    τ
                    l
                  
                
              
              ⁢
              
                
                  
                    
                      R
                      ~
                    
                    l
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                ⁢
                
                  
                    
                      S
                      ~
                    
                    k
                    *
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        l
                      
                    
                    )
                  
                
                ⁢
                
                  d
                  t
                
              
            
            =
            
              
                ∫
                0
                
                  T
                  s
                
              
              ⁢
              
                
                  
                    
                      R
                      ~
                    
                    l
                  
                  ⁡
                  
                    (
                    
                      t
                      +
                      
                        τ
                        l
                      
                    
                    )
                  
                
                ⁢
                
                  
                    
                      S
                      ~
                    
                    k
                    *
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                ⁢
                
                  d
                  t
                
              
            
          
        
      
    
    
      
        (
        7
        )
      
    
  




is the complex cross correlation of the lth received signal and the kth, signal waveform and





  
    
      
        
          E
          k
        
        =
        
          
            
              1
              2
            
            ⁢
            
              
                ∫
                0
                
                  T
                  s
                
              
              ⁢
              
                
                  
                    
                    
                      
                        
                          S
                          ~
                        
                        k
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    
                  
                  2
                
                ⁢
                
                  d
                  t
                
              
            
          
          =
          
            
              1
              2
            
            ⁢
            
              
                ∫
                
                  τ
                  l
                
                
                  
                    T
                    s
                  
                  +
                  
                    τ
                    l
                  
                
              
              ⁢
              
                
                  
                    
                    
                      
                        
                          S
                          ~
                        
                        k
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            l
                          
                        
                        )
                      
                    
                    
                  
                  2
                
                ⁢
                
                  d
                  t
                
              
            
          
        
      
    
    
      
        (
        8
        )
      
    
  




is the energy of the kth signal {tilde over (s)}sk(t). Furthermore, the constant K absorbs all the Kl.s as well as the factor





  
    e
    
      ∑
      
        l
        =
        1
      
      
        L
        p
      
    
  
  ⁢
  
    1
    
      2
      ⁢
      
        N
        i
      
    
  
  ⁢
  
    
      ∫
      0
      
        T
        S
      
    
    ⁢
    
      
        
          
          
            
              R
              l
            
            ⁡
            
              (
              t
              )
            
          
          
        
        2
      
      ⁢
      
        
          d
          t
        
        .
      
    
  




which is independent of K and thus has no bearing on the decision. Since, the natural logarithm is a monotonic function of its argument, we can equivalently maximize with respect to k and obtain (9) as follows:





  
    
      
        
          
            
              
                Λ
                k
              
              =
              
              ⁢
              
                ln
                ⁡
                
                  [
                  
                    p
                    ⁡
                    
                      (
                      
                        
                          
                            
                              {
                              
                                
                                  r
                                  l
                                
                                ⁡
                                
                                  (
                                  t
                                  )
                                
                              
                              }
                            
                            
                              l
                              =
                              1
                            
                            Lp
                          
                          |
                          
                            
                              s
                              k
                            
                            ⁡
                            
                              (
                              t
                              )
                            
                          
                        
                        ,
                        
                          
                            {
                            
                              α
                              l
                            
                            }
                          
                          
                            l
                            =
                            1
                          
                          Lp
                        
                        ,
                        
                          
                            {
                            
                              θ
                              l
                            
                            }
                          
                          
                            l
                            =
                            1
                          
                          Lp
                        
                        ,
                        
                          
                            {
                            
                              τ
                              l
                            
                            }
                          
                          
                            l
                            =
                            1
                          
                          Lp
                        
                      
                      )
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    ∑
                    
                      l
                      =
                      1
                    
                    
                      L
                      p
                    
                  
                  ⁢
                  
                    Re
                    ⁡
                    
                      [
                      
                        
                          
                            α
                            l
                          
                          
                            N
                            l
                          
                        
                        ⁢
                        
                          e
                          
                            -
                            
                              jθ
                              l
                            
                          
                        
                        ⁢
                        
                          ykl
                          ⁡
                          
                            (
                            τ
                            )
                          
                        
                      
                      ]
                    
                  
                
                -
                
                  
                    
                      α
                      l
                      2
                    
                    ⁢
                    
                      E
                      k
                    
                  
                  
                    N
                    i
                  
                
              
            
          
        
      
    
    
      
        
          
            
              (
              9
              )
            
          
        
        
          
            
              
                
                  
                    
                  
                
              
              
                
                  
                    (
                    10
                    )
                  
                
              
            
          
        
      
    
  




where K is ignored due to its independence with k.

Example 2
Known Phases and Delays, Unknown Amplitudes; Non-Coherent Detection
When the amplitudes of the received data signal are unknown, then the conditional probability of equation (6), must be averaged over their Joint Probability Density Function (JPDF) to arrive at the decision metric. Assuming independent amplitudes with first-order PDFs {pαl(αl)}l=1Lp we obtain: equation (11) as follows:




  
    p
    (
    
      
        
          
            {
            
              
                r
                l
              
              ⁡
              
                (
                t
                )
              
            
            }
          
          
            l
            =
            1
          
          Lp
        
        |
        
          
            s
            k
          
          ⁡
          
            (
            t
            )
          
        
      
      ,
      
        
          {
          
            θ
            l
          
          }
        
        
          l
          =
          1
        
        Lp
      
      ,
      
        
          {
          
            τ
            l
          
          }
        
        
          l
          =
          1
        
        Lp
      
    
    )
  
  =
  
    K
    ⁢
    
      
        ∏
        
          l
          =
          1
        
        
          L
          p
        
      
      ⁢
      
        
      
      ⁢
      
        
          ∫
          0
          ∞
        
        ⁢
        
          
            e
            
              
                Re
                ⁡
                
                  [
                  
                    
                      
                        a
                        l
                      
                      
                        N
                        l
                      
                    
                    ⁢
                    
                      e
                      
                        -
                        
                          jθ
                          l
                        
                      
                    
                    ⁢
                    
                      
                        y
                        kl
                      
                      ⁡
                      
                        (
                        τ
                        )
                      
                    
                  
                  ]
                
              
              -
              
                
                  
                    α
                    l
                    2
                  
                  ⁢
                  
                    E
                    k
                  
                
                
                  N
                  i
                
              
            
          
          ⁢
          
            
              p
              
                α
                l
              
            
            ⁡
            
              (
              
                α
                l
              
              )
            
          
          ⁢
          
            d
            
              α
              l
            
          
        
      
    
  




According to the invention, the PDF of Suzuki distribution is expressed mathematically as:




  
    
      
        
          
            p
            
              a
              l
            
          
          ⁡
          
            (
            
              a
              l
            
            )
          
        
        =
        
          
            
              (
              
                5
                2
              
              )
            
            
              1
              
                8
                ⁢
                
                  σ
                  2
                
              
            
          
          ⁢
          
            
              
                a
                l
              
              ⁢
              
                e
                
                  -
                  
                    
                      μ
                      2
                    
                    
                      2
                      ⁢
                      
                        σ
                        2
                      
                    
                  
                
              
            
            
              σ
              ⁢
              
                
                  2
                  ⁢
                  n
                
              
            
          
          ⁢
          
            Γ
            ⁡
            
              (
              
                1
                -
                
                  1
                  
                    8
                    ⁢
                    
                      σ
                      2
                    
                  
                
                -
                
                  μ
                  
                    2
                    ⁢
                    
                      σ
                      2
                    
                  
                
              
              )
            
          
          ⁢
          
            
              (
              
                
                  -
                  
                    1
                    
                      8
                      ⁢
                      
                        σ
                        2
                      
                    
                  
                
                -
                
                  
                    a
                    l
                    2
                  
                  2
                
              
              )
            
            
              
                -
                1
              
              +
              
                1
                
                  8
                  ⁢
                  
                    σ
                    2
                  
                
              
              +
              
                μ
                
                  2
                  ⁢
                  
                    σ
                    2
                  
                
              
            
          
        
      
    
    
      
        (
        12
        )
      
    
  




where,

σ=the variance of Lognormal distribution

μ=mean value of lognormal distribution

Therefore, by inserting equation (12) into equation (11) we obtain analytically:




  
    
      
        
          p
          (
          
            
              
                
                  {
                  
                    
                      r
                      l
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  }
                
                
                  l
                  =
                  1
                
                Lp
              
              |
              
                
                  s
                  k
                
                ⁡
                
                  (
                  t
                  )
                
              
            
            ,
            
              
                {
                
                  θ
                  l
                
                }
              
              
                l
                =
                1
              
              Lp
            
            ,
            
              
                {
                
                  τ
                  l
                
                }
              
              
                l
                =
                1
              
              Lp
            
          
          )
        
        =
        
          K
          ⁢
          
            
              ∏
              
                l
                =
                1
              
              
                L
                p
              
            
            ⁢
            
              
            
            ⁢
            
              
                ∫
                0
                ∞
              
              ⁢
              
                
                  
                    
                      e
                      
                        
                          Re
                          ⁡
                          
                            [
                            
                              
                                
                                  a
                                  l
                                
                                
                                  N
                                  l
                                
                              
                              ⁢
                              
                                e
                                
                                  -
                                  
                                    jθ
                                    l
                                  
                                
                              
                              ⁢
                              
                                
                                  y
                                  kl
                                
                                ⁡
                                
                                  (
                                  τ
                                  )
                                
                              
                            
                            ]
                          
                        
                        -
                        
                          
                            
                              α
                              l
                              2
                            
                            ⁢
                            
                              E
                              k
                            
                          
                          
                            N
                            i
                          
                        
                      
                    
                    ⁡
                    
                      (
                      
                        5
                        2
                      
                      )
                    
                  
                  
                    1
                    
                      8
                      ⁢
                      
                        σ
                        2
                      
                    
                  
                
                ⁢
                
                  
                    
                      a
                      l
                    
                    ⁢
                    
                      e
                      
                        -
                        
                          
                            μ
                            2
                          
                          
                            2
                            ⁢
                            
                              σ
                              2
                            
                          
                        
                      
                    
                  
                  
                    σ
                    ⁢
                    
                      
                        2
                        ⁢
                        n
                      
                    
                  
                
                ⁢
                
                  Γ
                  ⁡
                  
                    (
                    
                      1
                      -
                      
                        1
                        
                          8
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                      -
                      
                        μ
                        
                          2
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                    
                    )
                  
                
                ⁢
                
                  
                    (
                    
                      
                        -
                        
                          1
                          
                            8
                            ⁢
                            
                              σ
                              2
                            
                          
                        
                      
                      -
                      
                        
                          a
                          l
                          2
                        
                        2
                      
                    
                    )
                  
                  
                    
                      -
                      1
                    
                    +
                    
                      1
                      
                        8
                        ⁢
                        
                          σ
                          2
                        
                      
                    
                    +
                    
                      μ
                      
                        2
                        ⁢
                        
                          σ
                          2
                        
                      
                    
                  
                
                ⁢
                
                  d
                  
                    a
                    l
                  
                
              
            
          
        
      
    
    
      
        (
        13
        )
      
    
  




The integral in (13) can not be solved in closed-form as currently expressed, mainly due to the summation term under the power term. By assuming that




  
    
      
        
          
            
              (
              
                
                  1
                  
                    8
                    ⁢
                    
                      σ
                      2
                    
                  
                
                +
                
                  
                    a
                    l
                    2
                  
                  2
                
              
              )
            
            ≈
            
              a
              l
              2
            
          
          ,
          
            0
            ≤
            
              α
              l
            
            <
            1
          
        
        ⁢
        
          

        
        ⁢
        and
      
    
    
      
        (
        14
        )
      
    
  
  
    
      
        
          
            (
            
              
                1
                
                  8
                  ⁢
                  
                    σ
                    2
                  
                
              
              +
              
                
                  a
                  l
                  2
                
                2
              
            
            )
          
          ≈
          
            
              a
              l
              2
            
            2
          
        
        ,
        
          
            a
            l
          
          ≥
          1
        
      
    
    
      
        (
        15
        )
      
    
  




Equation (13) can become respectively,




  
    
      
        
          p
          (
          
            
              
                
                  {
                  
                    
                      r
                      l
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  }
                
                
                  l
                  =
                  1
                
                Lp
              
              |
              
                
                  s
                  k
                
                ⁡
                
                  (
                  t
                  )
                
              
            
            ,
            
              
                {
                
                  θ
                  l
                
                }
              
              
                l
                =
                1
              
              Lp
            
            ,
            
              
                {
                
                  τ
                  l
                
                }
              
              
                l
                =
                1
              
              Lp
            
          
          )
        
        =
        
          K
          ⁢
          
            
              ∏
              
                l
                =
                1
              
              
                L
                p
              
            
            ⁢
            
              
            
            ⁢
            
              
                ∫
                0
                ∞
              
              ⁢
              
                
                  
                    
                      e
                      
                        
                          Re
                          ⁡
                          
                            [
                            
                              
                                
                                  a
                                  l
                                
                                
                                  N
                                  l
                                
                              
                              ⁢
                              
                                e
                                
                                  -
                                  
                                    jθ
                                    l
                                  
                                
                              
                              ⁢
                              
                                
                                  y
                                  kl
                                
                                ⁡
                                
                                  (
                                  τ
                                  )
                                
                              
                            
                            ]
                          
                        
                        -
                        
                          
                            
                              α
                              l
                              2
                            
                            ⁢
                            
                              E
                              k
                            
                          
                          
                            N
                            i
                          
                        
                      
                    
                    ⁡
                    
                      (
                      
                        5
                        2
                      
                      )
                    
                  
                  
                    1
                    
                      8
                      ⁢
                      
                        σ
                        2
                      
                    
                  
                
                ⁢
                
                  
                    
                      a
                      l
                    
                    ⁢
                    
                      e
                      
                        -
                        
                          
                            μ
                            2
                          
                          
                            2
                            ⁢
                            
                              σ
                              2
                            
                          
                        
                      
                    
                  
                  
                    σ
                    ⁢
                    
                      
                        2
                        ⁢
                        n
                      
                    
                  
                
                ⁢
                
                  Γ
                  ⁡
                  
                    (
                    
                      1
                      -
                      
                        1
                        
                          8
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                      -
                      
                        μ
                        
                          2
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                    
                    )
                  
                
                ⁢
                
                  
                    (
                    
                      -
                      
                        a
                        l
                        2
                      
                    
                    )
                  
                  
                    
                      -
                      1
                    
                    +
                    
                      1
                      
                        8
                        ⁢
                        
                          σ
                          2
                        
                      
                    
                    +
                    
                      μ
                      
                        2
                        ⁢
                        
                          σ
                          2
                        
                      
                    
                  
                
                ⁢
                
                  d
                  
                    a
                    l
                  
                
              
            
          
        
      
    
    
      
        (
        16
        )
      
    
  
  
    
      
        
          
        
        ⁢
        and
      
    
    
      
        
      
    
  
  
    
      
        
          p
          (
          
            
              
                
                  {
                  
                    
                      r
                      l
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  }
                
                
                  l
                  =
                  1
                
                Lp
              
              |
              
                
                  s
                  k
                
                ⁡
                
                  (
                  t
                  )
                
              
            
            ,
            
              
                {
                
                  θ
                  l
                
                }
              
              
                l
                =
                1
              
              Lp
            
            ,
            
              
                {
                
                  τ
                  l
                
                }
              
              
                l
                =
                1
              
              Lp
            
          
          )
        
        =
        
          K
          ⁢
          
            
              ∏
              
                l
                =
                1
              
              
                L
                p
              
            
            ⁢
            
              
            
            ⁢
            
              
                ∫
                0
                ∞
              
              ⁢
              
                
                  
                    
                      e
                      
                        
                          Re
                          ⁡
                          
                            [
                            
                              
                                
                                  a
                                  l
                                
                                
                                  N
                                  l
                                
                              
                              ⁢
                              
                                e
                                
                                  -
                                  
                                    jθ
                                    l
                                  
                                
                              
                              ⁢
                              
                                
                                  y
                                  kl
                                
                                ⁡
                                
                                  (
                                  τ
                                  )
                                
                              
                            
                            ]
                          
                        
                        -
                        
                          
                            
                              α
                              l
                              2
                            
                            ⁢
                            
                              E
                              k
                            
                          
                          
                            N
                            i
                          
                        
                      
                    
                    ⁡
                    
                      (
                      
                        5
                        2
                      
                      )
                    
                  
                  
                    1
                    
                      8
                      ⁢
                      
                        σ
                        2
                      
                    
                  
                
                ⁢
                
                  
                    
                      a
                      l
                    
                    ⁢
                    
                      e
                      
                        -
                        
                          
                            μ
                            2
                          
                          
                            2
                            ⁢
                            
                              σ
                              2
                            
                          
                        
                      
                    
                  
                  
                    σ
                    ⁢
                    
                      
                        2
                        ⁢
                        n
                      
                    
                  
                
                ⁢
                
                  Γ
                  ⁡
                  
                    (
                    
                      1
                      -
                      
                        1
                        
                          8
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                      -
                      
                        μ
                        
                          2
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                    
                    )
                  
                
                ⁢
                
                  
                    (
                    
                      -
                      
                        
                          a
                          l
                          2
                        
                        2
                      
                    
                    )
                  
                  
                    
                      -
                      1
                    
                    +
                    
                      1
                      
                        8
                        ⁢
                        
                          σ
                          2
                        
                      
                    
                    +
                    
                      μ
                      
                        2
                        ⁢
                        
                          σ
                          2
                        
                      
                    
                  
                
                ⁢
                
                  d
                  
                    a
                    l
                  
                
              
            
          
        
      
    
    
      
        (
        17
        )
      
    
  




The closed-form solution of each of the above two equations are given below as follows




  
    
      
        
          p
          ⁡
          
            (
            
              
                
                  
                    {
                    
                      
                        r
                        l
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    }
                  
                  
                    l
                    =
                    1
                  
                  Lp
                
                |
                
                  
                    s
                    k
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
              ,
              
                
                  {
                  
                    θ
                    l
                  
                  }
                
                
                  l
                  =
                  1
                
                Lp
              
              ,
              
                
                  {
                  
                    τ
                    l
                  
                  }
                
                
                  l
                  =
                  1
                
                Lp
              
            
            )
          
        
        =
        
          
            K
            ⁢
            
              
                ∏
                
                  l
                  =
                  1
                
                
                  L
                  p
                
              
              ⁢
              
                
                  
                    (
                    
                      5
                      2
                    
                    )
                  
                  
                    1
                    
                      8
                      ⁢
                      
                        σ
                        2
                      
                    
                  
                
                ⁢
                
                  
                    e
                    
                      -
                      
                        
                          μ
                          2
                        
                        
                          2
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                    
                  
                  
                    2
                    ⁢
                    σ
                    ⁢
                    
                      
                        2
                        ⁢
                        n
                      
                    
                  
                
                ⁢
                
                  
                    
                      Γ
                      ⁡
                      
                        (
                        
                          1
                          -
                          
                            1
                            
                              8
                              ⁢
                              
                                σ
                                2
                              
                            
                          
                          -
                          
                            μ
                            
                              2
                              ⁢
                              
                                σ
                                2
                              
                            
                          
                        
                        )
                      
                    
                    ⁡
                    
                      [
                      
                        
                          N
                          l
                        
                        
                          E
                          k
                        
                      
                      ]
                    
                  
                  
                    
                      -
                      1
                    
                    +
                    
                      
                        1
                        +
                        
                          4
                          ⁢
                          μ
                        
                      
                      
                        4
                        ⁢
                        
                          σ
                          2
                        
                      
                    
                  
                
                ⁢
                
                  Γ
                  ⁡
                  
                    (
                    
                      
                        
                          1
                          +
                          
                            4
                            ⁢
                            μ
                          
                        
                        
                          8
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                      -
                      
                        1
                        2
                      
                    
                    )
                  
                
                ⁢
                
                  *
                  1
                
                ⁢
                
                  
                    F
                    1
                  
                  (
                  
                    
                      
                        
                          1
                          +
                          
                            4
                            ⁢
                            μ
                          
                        
                        
                          8
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                      -
                      
                        1
                        2
                      
                    
                    ,
                    
                      1
                      2
                    
                    ,
                    
                      
                        
                          e
                          
                            -
                            
                              j2θ
                              l
                            
                          
                        
                        ⁢
                        
                          
                            y
                            kl
                            2
                          
                          ⁡
                          
                            (
                            T
                            )
                          
                        
                      
                      
                        4
                        ⁢
                        
                          E
                          k
                        
                        ⁢
                        
                          N
                          l
                        
                      
                    
                  
                  )
                
              
            
          
          +
          
            
              [
              
                
              
              ⁢
              
                
                  
                    e
                    
                      -
                      
                        jθ
                        l
                      
                    
                  
                  ⁢
                  
                    
                      y
                      kl
                    
                    ⁡
                    
                      (
                      T
                      )
                    
                  
                
                
                  N
                  l
                
              
              ]
            
            ⁢
            
              
                
                  N
                  l
                
                
                  E
                  k
                
              
            
            ⁢
            
              
                Γ
                ⁡
                
                  (
                  
                    
                      1
                      +
                      
                        4
                        ⁢
                        μ
                      
                    
                    
                      8
                      ⁢
                      
                        σ
                        2
                      
                    
                  
                  )
                
              
              1
            
            ⁢
            
              
                F
                1
              
              (
              
                
                  
                    1
                    +
                    
                      4
                      ⁢
                      μ
                    
                  
                  
                    8
                    ⁢
                    
                      σ
                      2
                    
                  
                
                ,
                
                  3
                  2
                
                ,
                
                  
                    
                      e
                      
                        -
                        
                          j2θ
                          l
                        
                      
                    
                    ⁢
                    
                      
                        y
                        ki
                        2
                      
                      ⁡
                      
                        (
                        T
                        )
                      
                    
                  
                  
                    4
                    ⁢
                    
                      E
                      k
                    
                    ⁢
                    
                      N
                      l
                    
                  
                
              
              )
            
          
        
      
    
    
      
        (
        18
        )
      
    
  
  
    
      
        
          
        
        ⁢
        and
      
    
    
      
        
      
    
  
  
    
      
        
          p
          ⁢
          
            (
            
              
                
                  
                    {
                    
                      
                        r
                        l
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    }
                  
                  
                    l
                    =
                    1
                  
                  Lp
                
                |
                
                  
                    s
                    k
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
              ,
              
                
                  {
                  
                    θ
                    l
                  
                  }
                
                
                  l
                  =
                  1
                
                Lp
              
              ,
              
                
                  {
                  
                    τ
                    l
                  
                  }
                
                
                  l
                  =
                  1
                
                Lp
              
            
            )
          
        
        =
        
          
            K
            ⁢
            
              
                ∏
                
                  l
                  =
                  1
                
                
                  L
                  p
                
              
              ⁢
              
                
                  
                    (
                    
                      5
                      2
                    
                    )
                  
                  
                    1
                    
                      8
                      ⁢
                      
                        σ
                        2
                      
                    
                  
                
                ⁢
                
                  
                    e
                    
                      -
                      
                        
                          μ
                          2
                        
                        
                          2
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                    
                  
                  
                    2
                    ⁢
                    σ
                    ⁢
                    
                      
                        2
                        ⁢
                        n
                      
                    
                  
                
                ⁢
                
                  
                    
                      Γ
                      ⁡
                      
                        (
                        
                          1
                          -
                          
                            1
                            
                              8
                              ⁢
                              
                                σ
                                2
                              
                            
                          
                          -
                          
                            μ
                            
                              2
                              ⁢
                              
                                σ
                                2
                              
                            
                          
                        
                        )
                      
                    
                    ⁡
                    
                      [
                      
                        
                          N
                          l
                        
                        
                          E
                          k
                        
                      
                      ]
                    
                  
                  
                    
                      -
                      1
                    
                    +
                    
                      
                        1
                        +
                        
                          4
                          ⁢
                          μ
                        
                      
                      
                        4
                        ⁢
                        
                          σ
                          2
                        
                      
                    
                  
                
                ⁢
                
                  Γ
                  ⁡
                  
                    (
                    
                      
                        
                          1
                          +
                          
                            4
                            ⁢
                            μ
                          
                        
                        
                          8
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                      -
                      
                        1
                        2
                      
                    
                    )
                  
                
                *
                
                  
                    2
                    
                      1
                      -
                      
                        1
                        
                          8
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                      -
                      
                        μ
                        
                          2
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                    
                  
                  1
                
                ⁢
                
                  
                    F
                    1
                  
                  (
                  
                    
                      
                        
                          1
                          +
                          
                            4
                            ⁢
                            μ
                          
                        
                        
                          8
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                      -
                      
                        1
                        2
                      
                    
                    ,
                    
                      1
                      2
                    
                    ,
                    
                      
                        
                          e
                          
                            -
                            
                              j2θ
                              l
                            
                          
                        
                        ⁢
                        
                          
                            y
                            kl
                            2
                          
                          ⁡
                          
                            (
                            T
                            )
                          
                        
                      
                      
                        4
                        ⁢
                        
                          E
                          k
                        
                        ⁢
                        
                          N
                          l
                        
                      
                    
                  
                  )
                
              
            
          
          +
          
            
              [
              
                
                  
                    e
                    
                      -
                      
                        jθ
                        l
                      
                    
                  
                  ⁢
                  
                    
                      y
                      kl
                    
                    ⁡
                    
                      (
                      T
                      )
                    
                  
                
                
                  N
                  l
                
              
              ]
            
            ⁢
            
              
                
                  N
                  l
                
                
                  E
                  k
                
              
            
            ⁢
            
              
                Γ
                ⁡
                
                  (
                  
                    
                      1
                      +
                      
                        4
                        ⁢
                        μ
                      
                    
                    
                      8
                      ⁢
                      
                        σ
                        2
                      
                    
                  
                  )
                
              
              1
            
            ⁢
            
              
                F
                1
              
              (
              
                
                  
                    1
                    +
                    
                      4
                      ⁢
                      μ
                    
                  
                  
                    8
                    ⁢
                    
                      σ
                      2
                    
                  
                
                ,
                
                  3
                  2
                
                ,
                
                  
                    
                      e
                      
                        -
                        
                          j2θ
                          l
                        
                      
                    
                    ⁢
                    
                      
                        y
                        kl
                        2
                      
                      ⁡
                      
                        (
                        T
                        )
                      
                    
                  
                  
                    4
                    ⁢
                    
                      E
                      k
                    
                    ⁢
                    
                      N
                      l
                    
                  
                
              
              )
            
          
        
      
    
    
      
        (
        19
        )
      
    
  




Having managed to obtain (13) in a closed-form, the natural logarithm of it is taken in order to derive the decision metric of a receiver that has knowledge of the delays and phases of a Suzuki fading channel but not amplitudes. Therefore, one obtains:

Λk=ln [p({rl(t)}l=1Lp|sk(t),{θl}l=1Lp,{τl}l=1Lp)]  (20)

Based on that decision metric the receiver apparatus configuration design illustrated in FIG. 1 can be implemented for the case of known delay and phase values but unknown amplitude values for the data signal to be received. This design contains a block that theoretically includes the modulation schemes that can be possibly used. It also contains three different shaping pulses. Furthermore, non-coherent detection as well as uniform PDF are assumed.
Example 3
Known Data Signal Phase Value, Unknown Data Signal Delay and Amplitude Values; Non-Coherent Detection
After deriving successfully the decision metric for the case of known phases and delays but unknown amplitudes, we can derive next the decision metric for the case of known phases but unknown delays and amplitudes. However, it should be mentioned here that in typical practical applications, the fact that the phases are known while the delays are unknown is considered unorthodox. However, in quite narrowband systems (like OFDM) recent developments can make the realisation of such schemes possible. This can actually happen by determining the delays by the phase difference. Therefore, this design will be implemented since such case may attract more attention in future applications.
It is obvious that the difference between the current and the previous scheme is the lack of knowledge of the path delays. Hence, one has to average over the statistics of both amplitude and time, i.e, pαl(α1) and pτl(τ1).
The former is expressed by equation (12) while the latter is expressed by Alouini—initially Charash—as:




  
    
      
        
          
        
        ⁢
        
          
            
              
                
                  p
                  
                    τ
                    l
                  
                
                ⁡
                
                  (
                  
                    τ
                    l
                  
                  )
                
              
              =
              
                1
                
                  B
                  -
                  A
                
              
            
            ,
            
              A
              <
              
                τ
                l
              
              <
              B
            
          
          ⁢
          
            

          
          ⁢
          
            
              p
              (
              
                
                  
                    
                      {
                      
                        
                          r
                          l
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      }
                    
                    
                      l
                      =
                      1
                    
                    Lp
                  
                  |
                  
                    
                      s
                      k
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                ,
                
                  
                    {
                    
                      θ
                      l
                    
                    }
                  
                  
                    l
                    =
                    1
                  
                  Lp
                
              
              )
            
            =
            
              
                K
                ⁢
                
                  
                    ∏
                    
                      l
                      =
                      1
                    
                    
                      L
                      p
                    
                  
                  ⁢
                  
                    
                      ∫
                      A
                      B
                    
                    ⁢
                    
                      
                        ∫
                        0
                        ∞
                      
                      ⁢
                      
                        
                          e
                          
                            
                              Re
                              ⁡
                              
                                [
                                
                                  
                                    
                                      a
                                      l
                                    
                                    
                                      N
                                      l
                                    
                                  
                                  ⁢
                                  
                                    e
                                    
                                      -
                                      
                                        jθ
                                        l
                                      
                                    
                                  
                                  ⁢
                                  
                                    
                                      y
                                      kl
                                    
                                    ⁡
                                    
                                      (
                                      τ
                                      )
                                    
                                  
                                
                                ]
                              
                            
                            -
                            
                              
                                
                                  α
                                  l
                                  2
                                
                                ⁢
                                
                                  E
                                  k
                                
                              
                              
                                N
                                i
                              
                            
                          
                        
                        ⁢
                        
                          
                            p
                            
                              α
                              l
                            
                          
                          ⁡
                          
                            (
                            
                              α
                              l
                            
                            )
                          
                        
                        ⁢
                        
                          
                            p
                            
                              τ
                              l
                            
                          
                          ⁡
                          
                            (
                            
                              τ
                              l
                            
                            )
                          
                        
                        ⁢
                        
                          d
                          
                            α
                            l
                          
                        
                        ⁢
                        
                          d
                          
                            τ
                            l
                          
                        
                      
                    
                  
                
              
              =
            
          
        
      
    
    
      
        (
        21
        )
      
    
  




Thus, the decision metric is expressed mathematically as follows:




  
    
      
        
          
            p
            ⁡
            
              (
              
                
                  
                    
                      {
                      
                        
                          r
                          l
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      }
                    
                    
                      l
                      =
                      1
                    
                    Lp
                  
                  |
                  
                    
                      s
                      k
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                ,
                
                  
                    {
                    
                      θ
                      l
                    
                    }
                  
                  
                    l
                    =
                    1
                  
                  Lp
                
              
              )
            
          
          =
          
            K
            ⁢
            
              
                ∏
                
                  l
                  =
                  1
                
                
                  L
                  p
                
              
              ⁢
              
                
                  ∫
                  A
                  B
                
                ⁢
                
                  
                    p
                    (
                    
                      
                        
                          
                            {
                            
                              
                                r
                                l
                              
                              ⁡
                              
                                (
                                t
                                )
                              
                            
                            }
                          
                          
                            l
                            =
                            1
                          
                          Lp
                        
                        |
                        
                          
                            s
                            k
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                      
                      ,
                      
                        
                          {
                          
                            θ
                            l
                          
                          }
                        
                        
                          l
                          =
                          1
                        
                        Lp
                      
                      ,
                      
                        
                          {
                          
                            τ
                            l
                          
                          }
                        
                        
                          l
                          =
                          1
                        
                        Lp
                      
                    
                    )
                  
                  ⁢
                  
                    d
                    
                      τ
                      l
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          Λ
          k
        
        =
        
        ⁢
        
          
            ln
            ⁡
            
              [
              
                p
                ⁡
                
                  (
                  
                    
                      
                        
                          {
                          
                            
                              r
                              l
                            
                            ⁡
                            
                              (
                              t
                              )
                            
                          
                          }
                        
                        
                          l
                          =
                          1
                        
                        Lp
                      
                      |
                      
                        
                          s
                          k
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    ,
                    
                      
                        {
                        
                          θ
                          l
                        
                        }
                      
                      
                        l
                        =
                        1
                      
                      Lp
                    
                  
                  )
                
              
              ]
            
          
          =
          
            ln
            ⁢
            
              
            
            [
            
              K
              ⁢
              
                
                  ∏
                  
                    l
                    =
                    1
                  
                  
                    L
                    p
                  
                
                ⁢
                
                  
                    ∫
                    A
                    B
                  
                  ⁢
                  
                    
                      p
                      ⁡
                      
                        (
                        
                          
                            
                              
                                {
                                
                                  
                                    r
                                    l
                                  
                                  ⁡
                                  
                                    (
                                    t
                                    )
                                  
                                
                                }
                              
                              
                                l
                                =
                                1
                              
                              Lp
                            
                            |
                            
                              
                                s
                                k
                              
                              ⁡
                              
                                (
                                t
                                )
                              
                            
                          
                          ,
                          
                            
                              {
                              
                                θ
                                l
                              
                              }
                            
                            
                              l
                              =
                              1
                            
                            Lp
                          
                          ,
                          
                            
                              {
                              
                                τ
                                l
                              
                              }
                            
                            
                              l
                              =
                              1
                            
                            Lp
                          
                        
                        )
                      
                    
                    ⁢
                    
                      d
                      
                        τ
                        l
                      
                    
                  
                
              
            
            ]
          
        
      
    
    
      
        (
        22
        )
      
    
  




where the term





  1
  
    B
    -
    A
  




from pτl(τl) is absorbed within the constant ‘K’.

Therefore, it is straightforward to obtain the closed form expression for the decision metric:




  
    
      
        
          
            
              
                
                  Λ
                  k
                
                =
                
                ⁢
                
                  ln
                  ⁡
                  
                    [
                    
                      p
                      ⁡
                      
                        (
                        
                          
                            
                              
                                {
                                
                                  
                                    r
                                    l
                                  
                                  ⁡
                                  
                                    (
                                    t
                                    )
                                  
                                
                                }
                              
                              
                                l
                                =
                                1
                              
                              Lp
                            
                            |
                            
                              
                                s
                                k
                              
                              ⁡
                              
                                (
                                t
                                )
                              
                            
                          
                          ,
                          
                            
                              {
                              
                                θ
                                l
                              
                              }
                            
                            
                              l
                              =
                              1
                            
                            Lp
                          
                        
                        )
                      
                    
                    ]
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  ln
                  ⁡
                  
                    (
                    
                      
                        
                          
                            
                              K
                              ⁡
                              
                                (
                                
                                  B
                                  -
                                  A
                                
                                )
                              
                            
                            ⁢
                            
                              
                                ∏
                                
                                  l
                                  =
                                  1
                                
                                
                                  L
                                  p
                                
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      5
                                      2
                                    
                                    )
                                  
                                  
                                    1
                                    
                                      8
                                      ⁢
                                      
                                        σ
                                        2
                                      
                                    
                                  
                                
                                ⁢
                                
                                  
                                    e
                                    
                                      -
                                      
                                        
                                          μ
                                          2
                                        
                                        
                                          2
                                          ⁢
                                          
                                            σ
                                            2
                                          
                                        
                                      
                                    
                                  
                                  
                                    2
                                    ⁢
                                    σ
                                    ⁢
                                    
                                      
                                        2
                                        ⁢
                                        n
                                      
                                    
                                  
                                
                                ⁢
                                
                                  
                                    
                                      Γ
                                      ⁡
                                      
                                        (
                                        
                                          1
                                          -
                                          
                                            1
                                            
                                              8
                                              ⁢
                                              
                                                σ
                                                2
                                              
                                            
                                          
                                          -
                                          
                                            μ
                                            
                                              2
                                              ⁢
                                              
                                                σ
                                                2
                                              
                                            
                                          
                                        
                                        )
                                      
                                    
                                    ⁡
                                    
                                      [
                                      
                                        
                                          N
                                          l
                                        
                                        
                                          E
                                          k
                                        
                                      
                                      ]
                                    
                                  
                                  
                                    
                                      -
                                      1
                                    
                                    +
                                    
                                      
                                        1
                                        +
                                        
                                          4
                                          ⁢
                                          μ
                                        
                                      
                                      
                                        4
                                        ⁢
                                        
                                          σ
                                          2
                                        
                                      
                                    
                                  
                                
                                ⁢
                                
                                  Γ
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          
                                            1
                                            +
                                            
                                              4
                                              ⁢
                                              μ
                                            
                                          
                                          
                                            8
                                            ⁢
                                            
                                              σ
                                              2
                                            
                                          
                                        
                                        --
                                      
                                      ⁢
                                      
                                        1
                                        2
                                      
                                    
                                    )
                                  
                                
                              
                            
                          
                        
                      
                      
                        
                          
                            
                              
                                
                                  
                                    
                                      F
                                      1
                                      
                                      
                                        1
                                        
                                    
                                    (
                                    
                                      
                                        
                                          
                                            1
                                            +
                                            
                                              4
                                              ⁢
                                              μ
                                            
                                          
                                          
                                            8
                                            ⁢
                                            
                                              σ
                                              2
                                            
                                          
                                        
                                        -
                                        
                                          1
                                          2
                                        
                                      
                                      ,
                                      
                                        1
                                        2
                                      
                                      ,
                                      
                                        
                                          
                                            e
                                            
                                              -
                                              
                                                j2θ
                                                l
                                              
                                            
                                          
                                          ⁢
                                          
                                            
                                              y
                                              kl
                                              2
                                            
                                            ⁡
                                            
                                              (
                                              T
                                              )
                                            
                                          
                                        
                                        
                                          4
                                          ⁢
                                          
                                            E
                                            k
                                          
                                          ⁢
                                          
                                            N
                                            l
                                          
                                        
                                      
                                    
                                    )
                                  
                                  +
                                
                              
                            
                            
                              
                                
                                  
                                    [
                                    
                                      
                                        
                                          e
                                          
                                            -
                                            
                                              jθ
                                              l
                                            
                                          
                                        
                                        ⁢
                                        
                                          
                                            y
                                            kl
                                          
                                          ⁡
                                          
                                            (
                                            T
                                            )
                                          
                                        
                                      
                                      
                                        N
                                        l
                                      
                                    
                                    ]
                                  
                                  ⁢
                                  
                                    
                                      
                                        N
                                        l
                                      
                                      
                                        E
                                        k
                                      
                                    
                                  
                                  ⁢
                                  
                                    
                                      Γ
                                      ⁡
                                      
                                        (
                                        
                                          
                                            1
                                            +
                                            
                                              4
                                              ⁢
                                              μ
                                            
                                          
                                          
                                            8
                                            ⁢
                                            
                                              σ
                                              2
                                            
                                          
                                        
                                        )
                                      
                                    
                                    1
                                  
                                  ⁢
                                  
                                    
                                      F
                                      1
                                    
                                    (
                                    
                                      
                                        
                                          1
                                          +
                                          
                                            4
                                            ⁢
                                            μ
                                          
                                        
                                        
                                          8
                                          ⁢
                                          
                                            σ
                                            2
                                          
                                        
                                      
                                      ,
                                      
                                        3
                                        2
                                      
                                      ,
                                      
                                        
                                          
                                            e
                                            
                                              -
                                              
                                                j2θ
                                                l
                                              
                                            
                                          
                                          ⁢
                                          
                                            
                                              y
                                              ki
                                              2
                                            
                                            ⁡
                                            
                                              (
                                              T
                                              )
                                            
                                          
                                        
                                        
                                          4
                                          ⁢
                                          
                                            E
                                            k
                                          
                                          ⁢
                                          
                                            N
                                            l
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                              
                            
                          
                        
                      
                    
                    )
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        and
      
    
    
      
        (
        23
        )
      
    
  
  
    
      
        
          
            
              
                Λ
                k
              
              =
              
              ⁢
              
                ln
                ⁡
                
                  [
                  
                    p
                    ⁡
                    
                      (
                      
                        
                          
                            
                              {
                              
                                
                                  r
                                  l
                                
                                ⁡
                                
                                  (
                                  t
                                  )
                                
                              
                              }
                            
                            
                              l
                              =
                              1
                            
                            Lp
                          
                          |
                          
                            
                              s
                              k
                            
                            ⁡
                            
                              (
                              t
                              )
                            
                          
                        
                        ,
                        
                          
                            {
                            
                              θ
                              l
                            
                            }
                          
                          
                            l
                            =
                            1
                          
                          Lp
                        
                      
                      )
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                ln
                ⁡
                
                  (
                  
                    
                      
                        
                          
                            K
                            ⁡
                            
                              (
                              
                                B
                                -
                                A
                              
                              )
                            
                          
                          ⁢
                          
                            
                              ∏
                              
                                l
                                =
                                1
                              
                              
                                L
                                p
                              
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    5
                                    2
                                  
                                  )
                                
                                
                                  1
                                  
                                    8
                                    ⁢
                                    
                                      σ
                                      2
                                    
                                  
                                
                              
                              ⁢
                              
                                
                                  e
                                  
                                    -
                                    
                                      
                                        μ
                                        2
                                      
                                      
                                        2
                                        ⁢
                                        
                                          σ
                                          2
                                        
                                      
                                    
                                  
                                
                                
                                  2
                                  ⁢
                                  σ
                                  ⁢
                                  
                                    
                                      2
                                      ⁢
                                      n
                                    
                                  
                                
                              
                              ⁢
                              
                                
                                  
                                    Γ
                                    ⁡
                                    
                                      (
                                      
                                        1
                                        -
                                        
                                          1
                                          
                                            8
                                            ⁢
                                            
                                              σ
                                              2
                                            
                                          
                                        
                                        -
                                        
                                          μ
                                          
                                            2
                                            ⁢
                                            
                                              σ
                                              2
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                  ⁡
                                  
                                    [
                                    
                                      
                                        N
                                        l
                                      
                                      
                                        E
                                        k
                                      
                                    
                                    ]
                                  
                                
                                
                                  
                                    -
                                    1
                                  
                                  +
                                  
                                    
                                      1
                                      +
                                      
                                        4
                                        ⁢
                                        μ
                                      
                                    
                                    
                                      4
                                      ⁢
                                      
                                        σ
                                        2
                                      
                                    
                                  
                                
                              
                              ⁢
                              
                                Γ
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        
                                          1
                                          +
                                          
                                            4
                                            ⁢
                                            μ
                                          
                                        
                                        
                                          8
                                          ⁢
                                          
                                            σ
                                            2
                                          
                                        
                                      
                                      --
                                    
                                    ⁢
                                    
                                      1
                                      2
                                    
                                  
                                  )
                                
                              
                              *
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  
                                    
                                      2
                                      
                                        1
                                        -
                                        
                                          1
                                          
                                            8
                                            ⁢
                                            
                                              σ
                                              2
                                            
                                          
                                        
                                        -
                                        
                                          μ
                                          
                                            2
                                            ⁢
                                            
                                              σ
                                              2
                                            
                                          
                                        
                                      
                                    
                                  
                                
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                          
                                          1
                                        
                                        ⁢
                                        
                                          
                                            F
                                            1
                                          
                                          (
                                          
                                            
                                              
                                                
                                                  1
                                                  +
                                                  
                                                    4
                                                    ⁢
                                                    μ
                                                  
                                                
                                                
                                                  8
                                                  ⁢
                                                  
                                                    σ
                                                    2
                                                  
                                                
                                              
                                              -
                                              
                                                1
                                                2
                                              
                                            
                                            ,
                                            
                                              1
                                              2
                                            
                                            ,
                                            
                                              
                                                
                                                  e
                                                  
                                                    -
                                                    
                                                      j2θ
                                                      l
                                                    
                                                  
                                                
                                                ⁢
                                                
                                                  
                                                    y
                                                    kl
                                                    2
                                                  
                                                  ⁡
                                                  
                                                    (
                                                    T
                                                    )
                                                  
                                                
                                              
                                              
                                                4
                                                ⁢
                                                
                                                  E
                                                  k
                                                
                                                ⁢
                                                
                                                  N
                                                  l
                                                
                                              
                                            
                                          
                                          )
                                        
                                      
                                      +
                                    
                                  
                                
                              
                            
                          
                          
                            
                              
                                
                                  [
                                  
                                    
                                      
                                        e
                                        
                                          -
                                          
                                            jθ
                                            l
                                          
                                        
                                      
                                      ⁢
                                      
                                        
                                          y
                                          kl
                                        
                                        ⁡
                                        
                                          (
                                          T
                                          )
                                        
                                      
                                    
                                    
                                      N
                                      l
                                    
                                  
                                  ]
                                
                                ⁢
                                
                                  
                                    
                                      N
                                      l
                                    
                                    
                                      E
                                      k
                                    
                                  
                                
                                ⁢
                                
                                  
                                    Γ
                                    ⁡
                                    
                                      (
                                      
                                        
                                          1
                                          +
                                          
                                            4
                                            ⁢
                                            μ
                                          
                                        
                                        
                                          8
                                          ⁢
                                          
                                            σ
                                            2
                                          
                                        
                                      
                                      )
                                    
                                  
                                  1
                                
                                ⁢
                                
                                  
                                    F
                                    1
                                  
                                  (
                                  
                                    
                                      
                                        1
                                        +
                                        
                                          4
                                          ⁢
                                          μ
                                        
                                      
                                      
                                        8
                                        ⁢
                                        
                                          σ
                                          2
                                        
                                      
                                    
                                    ,
                                    
                                      3
                                      2
                                    
                                    ,
                                    
                                      
                                        
                                          e
                                          
                                            -
                                            
                                              j2θ
                                              l
                                            
                                          
                                        
                                        ⁢
                                        
                                          
                                            y
                                            ki
                                            2
                                          
                                          ⁡
                                          
                                            (
                                            T
                                            )
                                          
                                        
                                      
                                      
                                        4
                                        ⁢
                                        
                                          E
                                          k
                                        
                                        ⁢
                                        
                                          N
                                          l
                                        
                                      
                                    
                                  
                                  )
                                
                              
                            
                          
                        
                      
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        24
        )
      
    
  




Based on the decision metric expressed by equations (23) and (24), the receiver design illustrated in FIG. 2 can be implemented for the case of known phases but unknown delays and amplitudes. As in the above case, this design contains a block that theoretically includes the modulation schemes that can be possibly used together with the shaping pulses. Furthermore, non-coherent detection as well as uniform power decay profile (PDP) have been assumed.
Example 4
Known Delay and Amplitude Values, Unknown Phase Value; Non-Coherent Detection
In the case of known delay and amplitude values but unknown phase value the scheme is independent of the novel expression for the PDF of Suzuki distribution. However, the analysis of this scheme is vital since the derived results are used in the derivation process of any scheme that the phases are considered unknown.
When the phases are unknown, then the conditional probability p({rl(t)}l=1Lp|sk(t),{αl}l=1Lp,{τl}l=1Lp) has to be averaged over their joint probability density function (JPDF) in order to arrive at the decision metric. Assuming independent phases with PDFs over the interval {0, 2π}, one obtains:




  
    
      
        
          p
          ⁡
          
            (
            
              
                
                  
                    {
                    
                      
                        r
                        l
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    }
                  
                  
                    l
                    =
                    1
                  
                  Lp
                
                |
                
                  
                    s
                    k
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
              ,
              
                
                  {
                  
                    α
                    l
                  
                  }
                
                
                  l
                  =
                  1
                
                Lp
              
              ,
              
                
                  {
                  
                    τ
                    l
                  
                  }
                
                
                  l
                  =
                  1
                
                Lp
              
            
            )
          
        
        =
        
          
            K
            ⁢
            
              
                ∏
                
                  l
                  =
                  1
                
                
                  L
                  p
                
              
              ⁢
              
                
                  ∫
                  0
                  
                    2
                    ⁢
                    π
                  
                
                ⁢
                
                  
                    e
                    
                      
                        Re
                        ⁡
                        
                          [
                          
                            
                              
                                a
                                l
                              
                              
                                N
                                l
                              
                            
                            ⁢
                            
                              e
                              
                                -
                                
                                  jθ
                                  l
                                
                              
                            
                            ⁢
                            
                              
                                y
                                kl
                              
                              ⁡
                              
                                (
                                τ
                                )
                              
                            
                          
                          ]
                        
                      
                      -
                      
                        
                          
                            α
                            l
                            2
                          
                          ⁢
                          
                            E
                            k
                          
                        
                        
                          N
                          i
                        
                      
                    
                  
                  ⁢
                  
                    
                      p
                      
                        θ
                        l
                      
                    
                    ⁡
                    
                      (
                      
                        θ
                        l
                      
                      )
                    
                  
                  ⁢
                  
                    d
                    
                      θ
                      l
                    
                  
                
              
            
          
          =
        
      
    
    
      
        (
        25
        )
      
    
  
  
    
      
        =
        
          
            K
            ⁢
            
              
                ∏
                
                  l
                  =
                  1
                
                
                  L
                  p
                
              
              ⁢
              
                
                  e
                  
                    -
                    
                      
                        
                          α
                          l
                          2
                        
                        ⁢
                        
                          E
                          k
                        
                      
                      
                        N
                        i
                      
                    
                  
                
                ⁢
                
                  1
                  
                    2
                    ⁢
                    π
                  
                
                ⁢
                
                  
                    ∫
                    0
                    
                      2
                      ⁢
                      π
                    
                  
                  ⁢
                  
                    
                      e
                      
                        Re
                        ⁡
                        
                          [
                          
                            
                              
                                a
                                l
                              
                              
                                N
                                l
                              
                            
                            ⁢
                            
                              e
                              
                                -
                                
                                  jθ
                                  l
                                
                              
                            
                            ⁢
                            
                              
                                y
                                kl
                              
                              ⁡
                              
                                (
                                τ
                                )
                              
                            
                          
                          ]
                        
                      
                    
                    ⁢
                    
                      d
                      
                        θ
                        l
                      
                    
                  
                
              
            
          
          =
        
      
    
    
      
        (
        26
        )
      
    
  
  
    
      
        =
        
          
            K
            ⁢
            
              
                ∏
                
                  l
                  =
                  1
                
                
                  L
                  p
                
              
              ⁢
              
                
                  e
                  
                    -
                    
                      
                        
                          α
                          l
                          2
                        
                        ⁢
                        
                          E
                          k
                        
                      
                      
                        N
                        i
                      
                    
                  
                
                ⁢
                
                  1
                  
                    2
                    ⁢
                    n
                  
                
                ⁢
                
                  
                    ∫
                    0
                    
                      2
                      ⁢
                      π
                    
                  
                  ⁢
                  
                    
                      e
                      
                        Re
                        [
                        
                          
                            
                              a
                              l
                            
                            
                              N
                              l
                            
                          
                          ⁢
                          
                            
                              y
                              kl
                            
                            ⁡
                            
                              (
                              T
                              )
                            
                          
                          ⁢
                          
                            
                            
                              cos
                              (
                              
                                
                                  θ
                                  l
                                
                                -
                                
                                  arg
                                  (
                                  
                                    
                                      y
                                      kl
                                    
                                    ⁡
                                    
                                      (
                                      T
                                      )
                                    
                                  
                                  ]
                                
                              
                              
                            
                            ]
                          
                        
                      
                    
                    ⁢
                    
                      d
                      
                        θ
                        l
                      
                    
                  
                
              
            
          
          ⇒
        
      
    
    
      
        (
        27
        )
      
    
  
  
    
      
        
          p
          (
          
            
              
                
                  {
                  
                    
                      r
                      l
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  }
                
                
                  l
                  =
                  1
                
                Lp
              
              |
              
                
                  s
                  k
                
                ⁡
                
                  (
                  t
                  )
                
              
            
            ,
            
              
                {
                
                  α
                  l
                
                }
              
              
                l
                =
                1
              
              Lp
            
            ,
            
              
                {
                
                  τ
                  l
                
                }
              
              
                l
                =
                1
              
              Lp
            
          
          )
        
        =
        
          
            K
            ⁢
            
              
                ∏
                
                  l
                  =
                  1
                
                
                  L
                  p
                
              
              ⁢
              
                
                  e
                  
                    -
                    
                      
                        
                          α
                          l
                          2
                        
                        ⁢
                        
                          E
                          k
                        
                      
                      
                        N
                        i
                      
                    
                  
                
                ⁢
                
                  I
                  o
                
                ⁢
                
                  (
                  
                    
                      
                        a
                        l
                      
                      
                        N
                        l
                      
                    
                    |
                    
                      
                        y
                        kl
                      
                      ⁡
                      
                        (
                        
                          τ
                          l
                        
                        )
                      
                    
                    |
                  
                  )
                
              
            
          
          ⇒
        
      
    
    
      
        (
        28
        )
      
    
  
  
    
      
        
          
            Λ
            
              k
              =
            
          
          ⁢
          
            ln
            ⁡
            
              [
              
                p
                (
                
                  
                    
                      
                        {
                        
                          
                            r
                            l
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        }
                      
                      
                        l
                        =
                        1
                      
                      Lp
                    
                    |
                    
                      
                        s
                        k
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  ,
                  
                    
                      {
                      
                        α
                        l
                      
                      }
                    
                    
                      l
                      =
                      1
                    
                    Lp
                  
                  ,
                  
                    
                      {
                      
                        τ
                        l
                      
                      }
                    
                    
                      l
                      =
                      1
                    
                    Lp
                  
                
                )
              
              ]
            
          
        
        =
        
          K
          ⁢
          
            
              ∏
              
                l
                =
                1
              
              
                L
                p
              
            
            ⁢
            
              ln
              [
              
                
                  e
                  
                    -
                    
                      
                        
                          α
                          l
                          2
                        
                        ⁢
                        
                          E
                          k
                        
                      
                      
                        N
                        i
                      
                    
                  
                
                ⁢
                
                  
                    I
                    o
                  
                  ⁡
                  
                    (
                    
                      
                        
                          a
                          l
                        
                        
                          N
                          l
                        
                      
                      |
                      
                        
                          y
                          kl
                        
                        ⁡
                        
                          (
                          
                            τ
                            l
                          
                          )
                        
                      
                      |
                    
                    )
                  
                
              
              ]
            
          
        
      
    
    
      
        (
        29
        )
      
    
  




Example 5
Known Delay Values Unknown Phase and Amplitude Values for a Received Data Signal; Non-Coherent Detection
The case of known delay values but unknown phase and amplitude values is perhaps the most critical design to be considered for wireless communication engineering applications and the appropriate configuration of the receiving apparatus for the data signals, and can be derived from the closed form solution for the decision metric Of the case of known delays and amplitude values but unknown phase values.
The equation (28) is averaged over the PDF of Suzuki distribution pαl(αl). In other words, the mathematical expression of the decision metric of this scheme is the following:




  
    
      
        
          p
          ⁢
          
            (
            
              
                
                  
                    {
                    
                      
                        r
                        l
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    }
                  
                  
                    l
                    =
                    1
                  
                  Lp
                
                |
                
                  
                    s
                    k
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
              ,
              
                
                  {
                  
                    τ
                    l
                  
                  }
                
                
                  l
                  =
                  1
                
                Lp
              
            
            )
          
        
        =
        
          
            K
            ⁢
            
              
                ∏
                
                  l
                  =
                  1
                
                
                  L
                  p
                
              
              ⁢
              
                
                  ∫
                  0
                  ∞
                
                ⁢
                
                  
                    ∫
                    0
                    
                      2
                      ⁢
                      π
                    
                  
                  ⁢
                  
                    
                      e
                      
                        
                          Re
                          ⁡
                          
                            [
                            
                              
                                
                                  a
                                  l
                                
                                
                                  N
                                  l
                                
                              
                              ⁢
                              
                                e
                                
                                  -
                                  
                                    jθ
                                    l
                                  
                                
                              
                              ⁢
                              
                                
                                  y
                                  kl
                                
                                ⁡
                                
                                  (
                                  τ
                                  )
                                
                              
                            
                            ]
                          
                        
                        -
                        
                          
                            
                              α
                              l
                              2
                            
                            ⁢
                            
                              E
                              k
                            
                          
                          
                            N
                            i
                          
                        
                      
                    
                    ⁢
                    
                      
                        p
                        
                          θ
                          l
                        
                      
                      ⁡
                      
                        (
                        
                          θ
                          l
                        
                        )
                      
                    
                    ⁢
                    
                      
                        p
                        
                          α
                          l
                        
                      
                      ⁡
                      
                        (
                        
                          α
                          l
                        
                        )
                      
                    
                    ⁢
                    
                      d
                      
                        θ
                        l
                      
                    
                    ⁢
                    
                      d
                      
                        α
                        l
                      
                    
                  
                
              
            
          
          =
        
      
    
    
      
        (
        30
        )
      
    
  
  
    
      
        
          p
          ⁡
          
            (
            
              
                
                  
                    {
                    
                      
                        r
                        l
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    }
                  
                  
                    l
                    =
                    1
                  
                  Lp
                
                |
                
                  
                    s
                    k
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
              ,
              
                
                  {
                  
                    τ
                    l
                  
                  }
                
                
                  l
                  =
                  1
                
                Lp
              
            
            )
          
        
        =
        
          
            K
            ⁢
            
              
                ∏
                
                  l
                  =
                  1
                
                
                  L
                  p
                
              
              ⁢
              
                
                  ∫
                  0
                  ∞
                
                ⁢
                
                  
                    p
                    ⁡
                    
                      (
                      
                        
                          
                            
                              
                                
                                  
                                    {
                                    
                                      
                                        r
                                        l
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    }
                                  
                                  
                                    l
                                    =
                                    1
                                  
                                  Lp
                                
                                |
                                
                                  
                                    s
                                    k
                                  
                                  ⁢
                                  
                                    (
                                    t
                                    )
                                  
                                
                              
                              ,
                            
                          
                        
                        
                          
                            
                              
                                
                                  {
                                  
                                    α
                                    l
                                  
                                  }
                                
                                
                                  l
                                  =
                                  1
                                
                                Lp
                              
                              ,
                              
                                
                                  {
                                  
                                    τ
                                    l
                                  
                                  }
                                
                                
                                  l
                                  =
                                  1
                                
                                Lp
                              
                            
                          
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      p
                      
                        α
                        l
                      
                    
                    ⁡
                    
                      (
                      
                        α
                        l
                      
                      )
                    
                  
                  ⁢
                  
                    d
                    
                      α
                      l
                    
                  
                
              
            
          
          =
        
      
    
    
      
        (
        31
        )
      
    
  
  
    
      
        
          p
          ⁡
          
            (
            
              
                
                  
                    {
                    
                      
                        r
                        l
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    }
                  
                  
                    l
                    =
                    1
                  
                  Lp
                
                |
                
                  
                    s
                    k
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
              ,
              
                
                  {
                  
                    τ
                    l
                  
                  }
                
                
                  l
                  =
                  1
                
                Lp
              
            
            )
          
        
        =
        
          
            K
            ⁢
            
              
                ∏
                
                  l
                  =
                  1
                
                
                  L
                  p
                
              
              ⁢
              
                
                  ∫
                  0
                  ∞
                
                ⁢
                
                  
                    e
                    
                      -
                      
                        
                          
                            α
                            l
                            2
                          
                          ⁢
                          
                            E
                            k
                          
                        
                        
                          N
                          i
                        
                      
                    
                  
                  ⁢
                  
                    
                      I
                      o
                    
                    ⁡
                    
                      (
                      
                        
                          
                            a
                            l
                          
                          
                            N
                            l
                          
                        
                        |
                        
                          
                            y
                            kl
                          
                          ⁡
                          
                            (
                            
                              τ
                              l
                            
                            )
                          
                        
                        |
                      
                      )
                    
                  
                  ⁢
                  
                    
                      p
                      
                        α
                        l
                      
                    
                    ⁡
                    
                      (
                      
                        α
                        l
                      
                      )
                    
                  
                  ⁢
                  
                    d
                    
                      α
                      l
                    
                  
                
              
            
          
          =
        
      
    
    
      
        (
        32
        )
      
    
  




and by inserting the equation (12) into equation (32) we obtain (33) as follows:





  
    
      
        
          p
          (
          
            
              
                
                  {
                  
                    
                      r
                      l
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  }
                
                
                  l
                  =
                  1
                
                Lp
              
              |
              
                
                  s
                  k
                
                ⁡
                
                  (
                  t
                  )
                
              
            
            ,
            
              
                {
                
                  τ
                  l
                
                }
              
              
                l
                =
                1
              
              Lp
            
          
          )
        
        =
        
          K
          ⁢
          
            
              ∏
              
                l
                =
                1
              
              
                L
                p
              
            
            ⁢
            
              
                ∫
                0
                ∞
              
              ⁢
              
                
                  e
                  
                    -
                    
                      
                        
                          α
                          l
                          2
                        
                        ⁢
                        
                          E
                          k
                        
                      
                      
                        N
                        i
                      
                    
                  
                
                ⁢
                
                  
                    I
                    o
                  
                  ⁡
                  
                    (
                    
                      
                        
                          a
                          l
                        
                        
                          N
                          l
                        
                      
                      |
                      
                        
                          y
                          kl
                        
                        ⁡
                        
                          (
                          
                            τ
                            l
                          
                          )
                        
                      
                      |
                    
                    )
                  
                
                ⁢
                
                  
                    (
                    
                      5
                      2
                    
                    )
                  
                  
                    1
                    
                      8
                      ⁢
                      
                        σ
                        2
                      
                    
                  
                
                ⁢
                
                  
                    
                      a
                      l
                    
                    ⁢
                    
                      e
                      
                        -
                        
                          
                            μ
                            2
                          
                          
                            2
                            ⁢
                            
                              σ
                              2
                            
                          
                        
                      
                    
                  
                  
                    σ
                    ⁢
                    
                      
                        2
                        ⁢
                        n
                      
                    
                  
                
                ⁢
                
                  Γ
                  ⁡
                  
                    (
                    
                      1
                      -
                      
                        1
                        
                          8
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                      -
                      
                        μ
                        
                          2
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                    
                    )
                  
                
                ⁢
                
                  
                    (
                    
                      
                        -
                        
                          1
                          
                            8
                            ⁢
                            
                              σ
                              2
                            
                          
                        
                      
                      -
                      
                        
                          ·
                          
                            a
                            l
                            2
                          
                        
                        2
                      
                    
                    )
                  
                  
                    
                      -
                      1
                    
                    +
                    
                      1
                      
                        8
                        ⁢
                        
                          σ
                          2
                        
                      
                    
                    +
                    
                      μ
                      
                        2
                        ⁢
                        
                          σ
                          2
                        
                      
                    
                  
                
                ⁢
                
                  d
                  
                    a
                    l
                  
                
              
            
          
        
      
    
    
      
        (
        33
        )
      
    
  




Equation (33) cannot be solved in closed-form as currently expressed. The additional factor of complexity in this case is the modified Bessel function.
In order to simplify the summation term under the power term of Suzuki PDF, two assumptions were made based on the value of α1. The range of values of each of these two assumptions quite coincide with the range of values of the asymptotes of the modified Bessel function. In more detail, the modified Bessel function can be also expressed as:




  
    
      
        
          
            I
            b
          
          ⁡
          
            (
            x
            )
          
        
        =
        
          {
          
            
              
                
                  
                    
                      1
                      
                        Γ
                        ⁡
                        
                          (
                          
                            β
                            +
                            1
                          
                          )
                        
                      
                    
                    ⁢
                    
                      
                        (
                        
                          x
                          2
                        
                        )
                      
                      b
                    
                  
                  ,
                
              
              
                
                  0
                  <
                  x
                  ⪡
                  
                    
                      b
                      +
                      1
                    
                  
                
              
            
            
              
                
                  
                    
                      1
                      
                        
                          2
                          ⁢
                          π
                          ⁢
                          
                            
                          
                          ⁢
                          x
                        
                      
                    
                    ⁢
                    
                      e
                      x
                    
                  
                  ,
                
              
              
                
                  x
                  ⪢
                  
                    
                    
                      
                        b
                        2
                      
                      -
                      
                        1
                        4
                      
                    
                    
                  
                
              
            
          
        
      
    
    
      
        (
        34
        )
      
    
  




Therefore, by taking into consideration equation (34) as well as the assumptions in (14) and (15), we obtain respectively,




  
    
      
        
          
            p
            ⁡
            
              (
              
                
                  
                    
                      {
                      
                        r
                        l
                        
                          (
                          t
                          )
                        
                      
                      }
                    
                    
                      l
                      =
                      1
                    
                    Lp
                  
                  ❘
                  
                    
                      s
                      k
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                ,
                
                  
                    {
                    
                      τ
                      l
                    
                    }
                  
                  
                    l
                    =
                    1
                  
                  Lp
                
              
              )
            
          
          =
          
            
              K
              ⁢
              
                
                  ∏
                  
                    l
                    =
                    1
                  
                  
                    L
                    p
                  
                
                ⁢
                
                  
                    ∫
                    0
                    ∞
                  
                  ⁢
                  
                    
                      
                        
                          e
                          
                            -
                            
                              
                                
                                  a
                                  l
                                  2
                                
                                ⁢
                                
                                  E
                                  k
                                
                              
                              
                                N
                                l
                              
                            
                          
                        
                        ⁡
                        
                          (
                          
                            5
                            2
                          
                          )
                        
                      
                      
                        1
                        
                          8
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                    
                    ⁢
                    
                      
                        
                          a
                          l
                        
                        ⁢
                        
                          e
                          
                            -
                            
                              
                                μ
                                2
                              
                              
                                2
                                ⁢
                                
                                  σ
                                  2
                                
                              
                            
                          
                        
                      
                      
                        σ
                        ⁢
                        
                          
                            2
                            ⁢
                            π
                          
                        
                      
                    
                    ⁢
                    
                      Γ
                      ⁡
                      
                        (
                        
                          1
                          -
                          
                            1
                            
                              8
                              ⁢
                              
                                σ
                                2
                              
                            
                          
                          -
                          
                            μ
                            
                              2
                              ⁢
                              
                                σ
                                2
                              
                            
                          
                        
                        )
                      
                    
                    ⁢
                    
                      
                        (
                        
                          a
                          l
                          2
                        
                        )
                      
                      
                        
                          -
                          1
                        
                        +
                        
                          1
                          
                            8
                            ⁢
                            
                              σ
                              2
                            
                          
                        
                        +
                        
                          μ
                          
                            2
                            ⁢
                            
                              σ
                              2
                            
                          
                        
                      
                    
                    ⁢
                    
                      d
                      
                        α
                        l
                      
                    
                  
                
              
            
            =
            
              

            
            ⁢
            
              
            
            ⁢
            and
          
        
        ⁢
        
          

        
        ⁢
        
          K
          ⁢
          
            
              ∏
              
                l
                =
                1
              
              
                L
                p
              
            
            ⁢
            
              
                ∫
                0
                ∞
              
              ⁢
              
                
                  
                    
                      e
                      
                        -
                        
                          
                            
                              a
                              l
                              2
                            
                            ⁢
                            
                              E
                              k
                            
                          
                          
                            N
                            l
                          
                        
                      
                    
                    ⁡
                    
                      (
                      
                        5
                        2
                      
                      )
                    
                  
                  
                    1
                    
                      8
                      ⁢
                      
                        σ
                        2
                      
                    
                  
                
                ⁢
                
                  
                    
                      a
                      l
                    
                    ⁢
                    
                      e
                      
                        -
                        
                          
                            μ
                            2
                          
                          
                            2
                            ⁢
                            
                              σ
                              2
                            
                          
                        
                      
                    
                  
                  
                    σ
                    ⁢
                    
                      
                        2
                        ⁢
                        π
                      
                    
                  
                
                ⁢
                
                  
                    e
                    
                      
                        
                          a
                          l
                        
                        
                          N
                          l
                        
                      
                      ⁢
                      
                        
                        
                          
                            y
                            kl
                          
                          ⁡
                          
                            (
                            
                              τ
                              l
                            
                            )
                          
                        
                        
                      
                    
                  
                  
                    
                      2
                      ⁢
                      π
                      ⁢
                      
                        
                          a
                          l
                        
                        
                          N
                          l
                        
                      
                      ⁢
                      
                        
                        
                          
                            y
                            kl
                          
                          ⁡
                          
                            (
                            
                              τ
                              l
                            
                            )
                          
                        
                        
                      
                    
                  
                
                ⁢
                Γ
                ⁢
                
                  (
                  
                    1
                    -
                    
                      1
                      
                        8
                        ⁢
                        
                          σ
                          2
                        
                      
                    
                    -
                    
                      μ
                      
                        2
                        ⁢
                        
                          σ
                          2
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    (
                    
                      
                        a
                        l
                        2
                      
                      2
                    
                    )
                  
                  
                    
                      -
                      1
                    
                    +
                    
                      1
                      
                        8
                        ⁢
                        
                          σ
                          2
                        
                      
                    
                    +
                    
                      μ
                      
                        2
                        ⁢
                        
                          σ
                          2
                        
                      
                    
                  
                
                ⁢
                
                  d
                  
                    α
                    l
                  
                
              
            
          
        
      
    
    
      
        (
        35
        )
      
    
  




which has the following closed-form solutions, i.e.





  
    
      
        
          
            p
            ⁡
            
              (
              
                
                  
                    
                      {
                      
                        
                          r
                          l
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      }
                    
                    
                      l
                      =
                      1
                    
                    Lp
                  
                  ❘
                  
                    
                      s
                      k
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                ,
                
                  
                    {
                    
                      τ
                      l
                    
                    }
                  
                  
                    l
                    =
                    1
                  
                  Lp
                
              
              )
            
          
          =
          
            K
            ⁢
            
              
                ∏
                
                  l
                  =
                  1
                
                
                  L
                  p
                
              
              ⁢
              
                
                  
                    (
                    
                      5
                      2
                    
                    )
                  
                  
                    1
                    
                      8
                      ⁢
                      
                        σ
                        2
                      
                    
                  
                
                ⁢
                
                  
                    e
                    
                      -
                      
                        
                          μ
                          2
                        
                        
                          2
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                    
                  
                  
                    σ
                    ⁢
                    
                      
                        2
                        ⁢
                        π
                      
                    
                  
                
                ⁢
                
                  Γ
                  ⁡
                  
                    (
                    
                      1
                      -
                      
                        1
                        
                          8
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                      -
                      
                        μ
                        
                          2
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                    
                    )
                  
                
                ⁢
                
                  
                    
                      1
                      2
                    
                    ⁡
                    
                      [
                      
                        
                          E
                          k
                        
                        
                          N
                          l
                        
                      
                      ]
                    
                  
                  
                    -
                    
                      
                        1
                        +
                        
                          4
                          ⁢
                          μ
                        
                      
                      
                        8
                        ⁢
                        
                          σ
                          2
                        
                      
                    
                  
                
                ⁢
                
                  Γ
                  ⁡
                  
                    (
                    
                      
                        1
                        +
                        
                          4
                          ⁢
                          μ
                        
                      
                      
                        8
                        ⁢
                        
                          σ
                          2
                        
                      
                    
                    )
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          

        
        ⁢
        
          K
          ⁢
          
            
              ∏
              
                l
                =
                1
              
              
                L
                p
              
            
            ⁢
            
              
                
                  (
                  
                    5
                    2
                  
                  )
                
                
                  1
                  
                    8
                    ⁢
                    
                      σ
                      2
                    
                  
                
              
              ⁢
              
                
                  e
                  
                    -
                    
                      
                        μ
                        2
                      
                      
                        2
                        ⁢
                        
                          σ
                          2
                        
                      
                    
                  
                
                
                  4
                  ⁢
                  σπ
                  ⁢
                  
                    2
                  
                
              
              ⁢
              Γ
              ⁢
              
                
                  
                    (
                    
                      1
                      -
                      
                        1
                        
                          8
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                      -
                      
                        μ
                        
                          2
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                    
                    )
                  
                  ⁡
                  
                    [
                    
                      
                        E
                        k
                      
                      
                        N
                        l
                      
                    
                    ]
                  
                
                
                  
                    1
                    4
                  
                  -
                  
                    
                      1
                      +
                      
                        4
                        ⁢
                        μ
                      
                    
                    
                      8
                      ⁢
                      
                        σ
                        2
                      
                    
                  
                
              
              ⁢
              
                2
                
                  
                    1
                    2
                  
                  -
                  
                    1
                    
                      8
                      ⁢
                      
                        σ
                        2
                      
                    
                  
                  -
                  
                    μ
                    
                      2
                      ⁢
                      
                        σ
                        2
                      
                    
                  
                
              
              ⁢
              
                
                  
                    
                    
                      
                        y
                        kl
                      
                      ⁡
                      
                        (
                        
                          τ
                          l
                        
                        )
                      
                    
                    
                  
                  
                    N
                    l
                  
                
              
              ⁢
              
                (
                
                  
                    
                      Γ
                      ⁡
                      
                        (
                        
                          
                            
                              1
                              +
                              
                                4
                                ⁢
                                μ
                              
                            
                            
                              8
                              ⁢
                              
                                σ
                                2
                              
                            
                          
                          -
                          
                            1
                            4
                          
                        
                        )
                      
                    
                    ⁢
                    
                      
                        F
                        1
                        
                        
                          1
                          
                      
                      ⁡
                      
                        (
                        
                          
                            
                              
                                1
                                +
                                
                                  4
                                  ⁢
                                  μ
                                
                              
                              
                                8
                                ⁢
                                
                                  σ
                                  2
                                
                              
                            
                            -
                            
                              1
                              4
                            
                          
                          ,
                          
                            1
                            2
                          
                          ,
                          
                            
                              
                                y
                                kl
                                2
                              
                              ⁡
                              
                                (
                                τ
                                )
                              
                            
                            
                              4
                              ⁢
                              
                                E
                                k
                              
                              ⁢
                              
                                N
                                l
                              
                            
                          
                        
                        )
                      
                    
                  
                  -
                  
                    
                      [
                      
                        
                          
                            y
                            kl
                          
                          ⁡
                          
                            (
                            τ
                            )
                          
                        
                        
                          
                            
                              E
                              k
                            
                            ⁢
                            
                              N
                              l
                            
                          
                        
                      
                      ]
                    
                    ⁢
                    
                      Γ
                      ⁡
                      
                        (
                        
                          
                            
                              1
                              +
                              
                                4
                                ⁢
                                μ
                              
                            
                            
                              8
                              ⁢
                              
                                σ
                                2
                              
                            
                          
                          +
                          
                            1
                            4
                          
                        
                        )
                      
                    
                    ⁢
                    
                      
                        F
                        1
                        
                        
                          1
                          
                      
                      ⁡
                      
                        (
                        
                          
                            
                              
                                1
                                +
                                
                                  4
                                  ⁢
                                  μ
                                
                              
                              
                                8
                                ⁢
                                
                                  σ
                                  2
                                
                              
                            
                            +
                            
                              1
                              4
                            
                          
                          ,
                          
                            3
                            2
                          
                          ,
                          
                            
                              
                                y
                                kl
                                2
                              
                              ⁡
                              
                                (
                                τ
                                )
                              
                            
                            
                              4
                              ⁢
                              
                                E
                                k
                              
                              ⁢
                              
                                N
                                l
                              
                            
                          
                        
                        )
                      
                    
                  
                
                )
              
            
          
        
      
    
    
      
        (
        36
        )
      
    
  




and by taking the natural logarithm of (36) we obtain the decision metric for the case of known delays but unknown amplitudes and phases as follows:

Λk=ln [p({rl(t)}l=1Lp|sk(t),{τl}l=1Lp)]  (37)

Based on that decision metric the receiver design illustrated in FIG. 3 can be implemented for the case of known delays but unknown phases and amplitudes. As in the above case, this design contains a block that theoretically includes the modulation schemes that can be possibly used together with shaping pulses. Furthermore, coherent detection as well as uniform PDP have been selected in this design.
Example 6
Unknown Phase, Delay and Amplitude Values for a Received Data Signal; Non-Coherent Detection
When all channel parameters are unknown, the conditional probability has to be averaged over the statistics of the amplitudes, phases and delays which all are assumed statistically independent. However, since the decision metric has been derived for the case of known delays but unknown amplitudes and phases, we only need to average this metric over the PDF of time. The same methodology was followed above in the case of known phases but unknown amplitudes and delays. Therefore, the following expression must be solved in closed-form:




  
    
      
        
          p
          ⁡
          
            (
            
              
                
                  {
                  
                    
                      r
                      l
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  }
                
                
                  l
                  =
                  1
                
                Lp
              
              ❘
              
                
                  s
                  k
                
                ⁡
                
                  (
                  t
                  )
                
              
            
            )
          
        
        =
        
          
            
              ∫
              A
              B
            
            ⁢
            
              
                p
                ⁡
                
                  (
                  
                    
                      
                        
                          {
                          
                            
                              r
                              l
                            
                            ⁡
                            
                              (
                              t
                              )
                            
                          
                          }
                        
                        
                          l
                          =
                          1
                        
                        Lp
                      
                      ❘
                      
                        
                          s
                          k
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    ,
                    
                      
                        {
                        
                          τ
                          l
                        
                        }
                      
                      
                        l
                        =
                        1
                      
                      Lp
                    
                  
                  )
                
              
              ⁢
              
                
                  p
                  
                    τ
                    ⁢
                    
                      
                    
                    ⁢
                    l
                  
                
                ⁡
                
                  (
                  
                    τ
                    ⁢
                    
                      
                    
                    ⁢
                    l
                  
                  )
                
              
              ⁢
              
                d
                τ
              
              ⁢
              
                
              
              ⁢
              l
            
          
          ⇒
        
      
    
    
      
        (
        38
        )
      
    
  
  
    
      
        
          p
          ⁡
          
            (
            
              
                
                  {
                  
                    
                      r
                      l
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  }
                
                
                  l
                  =
                  1
                
                Lp
              
              ❘
              
                
                  s
                  k
                
                ⁡
                
                  (
                  t
                  )
                
              
            
            )
          
        
        =
        
          
            
              p
              ⁡
              
                (
                
                  
                    
                      
                        {
                        
                          
                            r
                            l
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        }
                      
                      
                        l
                        =
                        1
                      
                      Lp
                    
                    ❘
                    
                      
                        s
                        k
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  ,
                  
                    
                      {
                      
                        τ
                        l
                      
                      }
                    
                    
                      l
                      =
                      1
                    
                    Lp
                  
                
                )
              
            
            ⁢
            
              
                ∫
                A
                B
              
              ⁢
              
                
                  
                    p
                    
                      τ
                      ⁢
                      
                        
                      
                      ⁢
                      l
                    
                  
                  ⁡
                  
                    (
                    
                      τ
                      ⁢
                      
                        
                      
                      ⁢
                      l
                    
                    )
                  
                
                ⁢
                
                  d
                  τ
                
                ⁢
                
                  
                
                ⁢
                l
              
            
          
          ⇒
        
      
    
    
      
        (
        39
        )
      
    
  
  
    
      
        
          
            Λ
            k
          
          =
          
            
              ln
              ⁡
              
                [
                
                  p
                  ⁡
                  
                    (
                    
                      
                        
                          {
                          
                            
                              r
                              l
                            
                            ⁡
                            
                              (
                              t
                              )
                            
                          
                          }
                        
                        
                          l
                          =
                          1
                        
                        Lp
                      
                      ❘
                      
                        
                          s
                          k
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    )
                  
                
                ]
              
            
            =
            
              
                ln
                [
                
                  
                    K
                    ⁡
                    
                      (
                      
                        B
                        -
                        A
                      
                      )
                    
                  
                  ⁢
                  
                    
                      ∏
                      
                        l
                        =
                        1
                      
                      
                        L
                        p
                      
                    
                    ⁢
                    
                      
                        
                          (
                          
                            5
                            2
                          
                          )
                        
                        
                          1
                          
                            8
                            ⁢
                            
                              σ
                              2
                            
                          
                        
                      
                      ⁢
                      
                        
                          e
                          
                            -
                            
                              
                                μ
                                2
                              
                              
                                2
                                ⁢
                                
                                  σ
                                  2
                                
                              
                            
                          
                        
                        
                          σ
                          ⁢
                          
                            
                              2
                              ⁢
                              π
                            
                          
                        
                      
                      ⁢
                      
                        Γ
                        ⁡
                        
                          (
                          
                            1
                            -
                            
                              1
                              
                                8
                                ⁢
                                
                                  σ
                                  2
                                
                              
                            
                            -
                            
                              μ
                              
                                2
                                ⁢
                                
                                  σ
                                  2
                                
                              
                            
                          
                          )
                        
                      
                    
                  
                
                ]
              
              ⁢
              
                
                  
                    1
                    2
                  
                  [
                  
                    
                      E
                      k
                    
                    
                      N
                      l
                    
                  
                  ]
                
                
                  -
                  
                    
                      1
                      +
                      
                        4
                        ⁢
                        μ
                      
                    
                    
                      8
                      ⁢
                      
                        σ
                        2
                      
                    
                  
                
              
              ⁢
              
                Γ
                ⁡
                
                  (
                  
                    
                      1
                      +
                      
                        4
                        ⁢
                        μ
                      
                    
                    
                      8
                      ⁢
                      
                        σ
                        2
                      
                    
                  
                  )
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          

        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            ln
            ⁡
            
              [
              
                
                  
                    
                      
                        K
                        ⁡
                        
                          (
                          
                            B
                            -
                            A
                          
                          )
                        
                      
                      ⁢
                      
                        
                          ∏
                          
                            l
                            =
                            1
                          
                          
                            L
                            p
                          
                        
                        ⁢
                        
                          
                            
                              (
                              
                                5
                                2
                              
                              )
                            
                            
                              1
                              
                                8
                                ⁢
                                
                                  σ
                                  2
                                
                              
                            
                          
                          ⁢
                          
                            
                              e
                              
                                -
                                
                                  
                                    μ
                                    2
                                  
                                  
                                    2
                                    ⁢
                                    
                                      σ
                                      2
                                    
                                  
                                
                              
                            
                            
                              4
                              ⁢
                              σπ
                              ⁢
                              
                                2
                              
                            
                          
                          ⁢
                          
                            
                              
                                Γ
                                ⁡
                                
                                  (
                                  
                                    1
                                    -
                                    
                                      1
                                      
                                        8
                                        ⁢
                                        
                                          σ
                                          2
                                        
                                      
                                    
                                    -
                                    
                                      μ
                                      
                                        2
                                        ⁢
                                        
                                          σ
                                          2
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              ⁡
                              
                                [
                                
                                  
                                    E
                                    k
                                  
                                  
                                    N
                                    l
                                  
                                
                                ]
                              
                            
                            
                              
                                1
                                4
                              
                              -
                              
                                
                                  1
                                  +
                                  
                                    4
                                    ⁢
                                    μ
                                  
                                
                                
                                  8
                                  ⁢
                                  
                                    σ
                                    2
                                  
                                
                              
                            
                          
                          ⁢
                          
                            2
                            
                              
                                1
                                2
                              
                              -
                              
                                1
                                
                                  8
                                  ⁢
                                  
                                    σ
                                    2
                                  
                                
                              
                              -
                              
                                
                                  μ
                                  
                                    2
                                    ⁢
                                    
                                      σ
                                      2
                                    
                                  
                                
                                ⁢
                                
                                  
                                    
                                      
                                      
                                        
                                          y
                                          kl
                                        
                                        ⁡
                                        
                                          (
                                          
                                            τ
                                            l
                                          
                                          )
                                        
                                      
                                      
                                    
                                    
                                      N
                                      l
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
                
                  
                    
                      (
                      
                        
                          Γ
                          ⁢
                          
                            (
                            
                              
                                
                                  1
                                  +
                                  
                                    4
                                    ⁢
                                    μ
                                  
                                
                                
                                  8
                                  ⁢
                                  
                                    σ
                                    2
                                  
                                
                              
                              -
                              
                                1
                                4
                              
                            
                            )
                          
                          ⁢
                          
                            
                              F
                              1
                              
                              
                                1
                                
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    
                                      1
                                      +
                                      
                                        4
                                        ⁢
                                        μ
                                      
                                    
                                    
                                      8
                                      ⁢
                                      
                                        σ
                                        2
                                      
                                    
                                  
                                  -
                                  
                                    1
                                    4
                                  
                                
                                ,
                                
                                  1
                                  2
                                
                                ,
                                
                                  
                                    
                                      y
                                      kl
                                      2
                                    
                                    ⁡
                                    
                                      (
                                      τ
                                      )
                                    
                                  
                                  
                                    4
                                    ⁢
                                    
                                      E
                                      k
                                    
                                    ⁢
                                    
                                      N
                                      l
                                    
                                  
                                
                              
                              )
                            
                          
                        
                        -
                        
                          
                            [
                            
                              
                                
                                  y
                                  kl
                                
                                ⁡
                                
                                  (
                                  τ
                                  )
                                
                              
                              
                                
                                  
                                    E
                                    k
                                  
                                  ⁢
                                  
                                    N
                                    l
                                  
                                
                              
                            
                            ]
                          
                          ⁢
                          
                            Γ
                            ⁡
                            
                              (
                              
                                
                                  
                                    1
                                    +
                                    
                                      4
                                      ⁢
                                      μ
                                    
                                  
                                  
                                    8
                                    ⁢
                                    
                                      σ
                                      2
                                    
                                  
                                
                                +
                                
                                  1
                                  4
                                
                              
                              )
                            
                          
                          ⁢
                          
                            
                              F
                              1
                              
                              
                                1
                                
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    
                                      1
                                      +
                                      
                                        4
                                        ⁢
                                        μ
                                      
                                    
                                    
                                      8
                                      ⁢
                                      
                                        σ
                                        2
                                      
                                    
                                  
                                  +
                                  
                                    1
                                    4
                                  
                                
                                ,
                                
                                  3
                                  2
                                
                                ,
                                
                                  
                                    
                                      y
                                      kl
                                      2
                                    
                                    ⁡
                                    
                                      (
                                      τ
                                      )
                                    
                                  
                                  
                                    4
                                    ⁢
                                    
                                      E
                                      k
                                    
                                    ⁢
                                    
                                      N
                                      l
                                    
                                  
                                
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
              
              ]
            
          
          .
        
      
    
    
      
        (
        40
        )
      
    
  




Based on this decision metric we can implement the receiver design as illustrated in FIG. 4 for the case of unknown delays, phases and amplitudes. As in the above case, this design contains a block that theoretically includes the modulation schemes that can be possibly used together with the shaping pulses. Furthermore, coherent detection as well as uniform PDP have been selected in this design.
Power Decay Profile
The Power Decay Profile (PDP) critically affects the performance of wireless communications systems and it is always considered as a factor of significant importance during the system design process. In the majority of systems, the PDP is typically considered to be flat i.e., Ωl=Ω; l=1, 2, . . . , Lp. The PDP is considered to be of the same nature in all diversity paths between transmitter and receiver.
Uniform PDP indicates the uniform rate of variation of the power of the signal while Exponential PDP indicates its exponential variation. The PDP is described mathematically as follows:

γl=γ1e−δ(l−1),l=1,2, . . . ,Lp  (41)

where γl is the signal-to-noise ratio in the lth diversity path—





  
    (
    
      SNR
      l
    
    )
  
  =
  
    
      
        E
        k
      
      
        N
        l
      
    
    ⁢
    
      -
    
  




and δ is the fading power decay profile factor which typically takes values between zero and two.

In the present case we will consider the values δ=0, 0.1, 0.5, 1.0, 2.0, while it is obvious that for δ=0, the exponential PDP becomes uniform PDP.
In addition, the nature of the PDP determines the degree of optimisation of a receiver design. For uniform PDP, the receiver design is considered optimum, while, for non-uniform PDP (exponential in our case), the receiver design is considered suboptimum.
Coherent Detection
Hereinabove it is assumed that the channel is constant for one symbol period i.e. the detection is considered non-coherent. However, it is well known that coherent detection is always desired ideally. Coherent detection assumes that the channel remains constant for at least two symbol periods. This can be realised mathematically by changing the upper limit of the integral in equation (7) from ‘Ts+τl’ to ‘2Ts+τl’. Therefore; only the term ‘ykl’ changes and varies proportionally the value decision metric. Hence, this small change of ‘Ts’ to ‘2Ts’ is the difference between the optimum/suboptimum receiver designs with non-coherent and coherent detection respectively.
Shaping Pulses
Using shaping pulses in the transmitter, the data signal which is transmitted can be shaped in a desired waveform which will Make it relatively easier for the receiver to filter the received signals effectively and extract the information included in it. The typical shaping pulses that are used are the square, sinusoidal and cosine. These three pulses are expressed mathematically, respectively, as follows:




  
    
      
        
          
            g
            ⁡
            
              (
              t
              )
            
          
          ⁢
          
            ❘
            r
          
        
        =
        
          
            
              2
              ⁢
              
                E
                b
              
            
            
              T
              b
            
          
        
      
    
    
      
        (
        42
        )
      
    
  
  
    
      
        
          
            g
            ⁡
            
              (
              t
              )
            
          
          ⁢
          
            ❘
            s
          
        
        =
        
          {
          
            
              
                
                  
                    
                      sin
                      ⁡
                      
                        (
                        
                          π
                          ⁢
                          
                            
                          
                          ⁢
                          τ
                        
                        )
                      
                    
                    
                      2
                      ⁢
                      T
                    
                  
                  ,
                
              
              
                
                  0
                  ≤
                  t
                  ≤
                  
                    2
                    ⁢
                    T
                  
                
              
            
            
              
                
                  0
                  ,
                
              
              
                otherwise
              
            
          
          }
        
      
    
    
      
        (
        43
        )
      
    
  
  
    
      
        
          
            g
            ⁡
            
              (
              t
              )
            
          
          ⁢
          
            ❘
            c
          
        
        =
        
          {
          
            
              
                
                  
                    
                      1
                      
                        2
                        ⁢
                        L
                        ⁢
                        
                          
                        
                        ⁢
                        T
                      
                    
                    ⁡
                    
                      [
                      
                        1
                        -
                        
                          
                            cos
                            ⁡
                            
                              (
                              
                                2
                                ⁢
                                πτ
                              
                              )
                            
                          
                          
                            L
                            ⁢
                            
                              
                            
                            ⁢
                            T
                          
                        
                      
                      ]
                    
                  
                  ,
                
              
              
                
                  0
                  ≤
                  t
                  ≤
                  
                    2
                    ⁢
                    T
                  
                
              
            
            
              
                
                  0
                  ,
                
              
              
                otherwise
              
            
          
          }
        
      
    
    
      
        (
        44
        )
      
    
  




where L=1, for full response and L>1 for partial response, result in smaller bandwidth efficiency, hence, greater bandwidth efficiency than the use of rectangular pulses.

Digital Modulation Schemes

Digital modulation techniques can be an important factor in the transmission of digital data in order to achieve the highest possible data transmission rates. The higher the order of the modulation, the higher the spectral efficiency i.e. the higher the number of bits per transmitted symbol. However, in high order modulation schemes the complexity, and hence, the cost increase dramatically. In addition, the information symbols are located very close to each other which results in inter symbol interference (ISO. This increases the error rate and raises the need for more sophisticated, complex and costly receiver designs. Therefore, the choice of the appropriate modulation scheme for each system is relative to the performance requirements of it. Six types of digital modulations, namely, M-PSK, M-QAM, ASK/PAM, FSK, MSK and OFDM can be expressed mathematically as follows:




  
    
      
        
          M
          ⁢
          
            -
          
          ⁢
          PSK
        
        ⁢
        
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                S
                k
              
              ⁡
              
                (
                t
                )
              
            
            ⁢
            
              ❘
              
                M
                ⁢
                
                  -
                
                ⁢
                PSK
              
            
          
          =
          
            
              
                g
                ⁡
                
                  (
                  t
                  )
                
              
              ⁢
              
                cos
                ⁡
                
                  [
                  
                    
                      2
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      
                        f
                        c
                      
                      ⁢
                      t
                    
                    +
                    
                      
                        
                          2
                          ⁢
                          π
                        
                        M
                      
                      ⁢
                      
                        (
                        
                          m
                          -
                          1
                        
                        )
                      
                    
                  
                  ]
                
              
              ⁢
              
                
              
              ⁢
              m
            
            =
            1
          
        
        ,
        2
        ,
        ...
        ⁢
        
          
        
        ,
        M
      
    
    
      
        (
        45
        )
      
    
  




where g(t) stands for the shaping pulse, fc is the carrier frequency, M is the number of symbol states





  
    
      
        M
        ⁢
        
          -
        
        ⁢
        QAM
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              S
              k
            
            ⁡
            
              (
              t
              )
            
          
          ⁢
          
            ❘
            
              M
              ⁢
              
                -
              
              ⁢
              QAM
            
          
        
        =
        
          
            
              Re
              ⁡
              
                [
                
                  
                    A
                    mc
                  
                  +
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    
                      A
                      ms
                    
                  
                
                ]
              
            
            ⁢
            
              g
              ⁡
              
                (
                t
                )
              
            
            ⁢
            
              e
              
                2
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                f
              
            
            ⁢
            
              c
              t
            
          
          =
        
      
    
    
      
        (
        46
        )
      
    
  
  
    
      
        =
        
          
            
              A
              mc
            
            ⁢
            
              g
              ⁡
              
                (
                t
                )
              
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  2
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  
                    f
                    c
                  
                  ⁢
                  t
                
                )
              
            
          
          -
          
            
              A
              ms
            
            ⁢
            
              g
              ⁡
              
                (
                t
                )
              
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  2
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  
                    f
                    c
                  
                  ⁢
                  t
                
                )
              
            
          
        
      
    
    
      
        (
        47
        )
      
    
  




where Amc and Ams stand for the mapping amplitudes in the real and imaginary axis, respectively.





  
    
      
        ASK
        ⁢
        
          /
        
        ⁢
        PAM
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              S
              k
            
            ⁡
            
              (
              t
              )
            
          
          ⁢
          
            ❘
            
              ASK
              /
              PAM
            
          
        
        =
        
          
            Re
            ⁡
            
              [
              
                
                  A
                  m
                
                ⁢
                
                  g
                  ⁡
                  
                    (
                    t
                    )
                  
                
                ⁢
                
                  e
                  
                    2
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    
                      f
                      c
                    
                    ⁢
                    t
                  
                
              
              ]
            
          
          =
        
      
    
    
      
        (
        48
        )
      
    
  
  
    
      
        
          =
          
            
              
                A
                m
              
              ⁢
              
                g
                ⁡
                
                  (
                  t
                  )
                
              
              ⁢
              
                cos
                ⁡
                
                  (
                  
                    2
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    
                      f
                      c
                    
                    ⁢
                    t
                  
                  )
                
              
              ⁢
              
                
              
              ⁢
              m
            
            =
            1
          
        
        ,
        2
        ,
        ...
        ⁢
        
          
        
        ,
        M
      
    
    
      
        (
        49
        )
      
    
  
  
    
      MSK
    
    
      
        
      
    
  
  
    
      
        
          
            
              S
              k
            
            ⁡
            
              (
              t
              )
            
          
          ⁢
          
            ❘
            MSK
          
        
        =
        
          
            
              A
              mc
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                  2
                
                )
              
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  2
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  
                    f
                    c
                  
                  ⁢
                  t
                
                )
              
            
          
          -
          
            
              A
              ms
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                  2
                
                )
              
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  2
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  
                    f
                    c
                  
                  ⁢
                  t
                
                )
              
            
          
        
      
    
    
      
        (
        50
        )
      
    
  
  
    
      FSK
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                S
                k
              
              ⁡
              
                (
                t
                )
              
            
            ⁢
            
              ❘
              FSK
            
          
          =
          
            
              
                g
                ⁡
                
                  (
                  t
                  )
                
              
              ⁢
              
                cos
                ⁡
                
                  [
                  
                    
                      (
                      
                        2
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        
                          f
                          c
                        
                        ⁢
                        t
                      
                      )
                    
                    +
                    
                      2
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      m
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                      ⁢
                      
                        
                      
                      ⁢
                      f
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                  
                  ]
                
              
              ⁢
              
                
              
              ⁢
              m
            
            =
            1
          
        
        ,
        2
        ,
        ...
        ⁢
        
          
        
        ,
        M
      
    
    
      
        (
        51
        )
      
    
  
  
    
      OFDM
    
    
      
        
      
    
  
  
    
      
        
          
            
              S
              k
            
            ⁡
            
              (
              t
              )
            
          
          ⁢
          
            |
            OFDM
          
        
        =
        
          
            1
            2
          
          ⁢
          
            
              ∑
              
                k
                =
                0
              
              
                N
                -
                1
              
            
            ⁢
            
              
                X
                k
              
              ⁢
              
                e
                
                  
                    j2π
                    ⁢
                    
                      
                    
                    ⁢
                    k
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                  T
                
              
              ⁢
              
                e
                
                  j2π
                  ⁢
                  
                    
                  
                  ⁢
                  
                    f
                    c
                  
                  ⁢
                  t
                
              
            
          
        
      
    
    
      
        (
        52
        )
      
    
  




where Xk is the kth transmitted symbol and T is the period of each OFDM symbol. 1/T stands for the subcarrier spacing and it is actually the factor that provides the orthogonality between the subcarriers consisted within each OFDM.

Thus in accordance with the invention the configuration design of the receiving apparatus can be achieved by referring to the appropriate decision metric. As indicated above and in FIGS. 1-4, four basic receiver designs are produced directly from the four different derived decision metrics in equations (20), (24), (37), (40).
However, where non-coherent detection is provided in these designs, four modified designs can be produced for the cases of coherent detection.
In addition, the PDP was considered uniform in the four initial designs. Hence receiver designs with coherent and non-coherent detection under exponential PDP can be produced in a straightforward manner. In addition, each of the designs can be changed to take into account the six modulation schemes, and different shaping pulses can be used in all schemes except MSK and OFDM. Therefore, the number of receiver configuration designs which can be achieved are:




  
    
      
        4
        ⁢
        
          (
          Initial
          )
        
        *
        4
        ⁢
        
          (
          modulations
          )
        
        *
        2
        ⁢
        
          (
          PDP
          )
        
        *
        2
        ⁢
        
          (
          detections
          )
        
        *
        3
        ⁢
        
          
            (
            pulses
            )
          
          ⁡
          
            [
            
              =
              192
            
            ]
          
        
      
      +
      
        4
        ⁢
        
          (
          Initial
          )
        
        *
        2
        ⁢
        
          (
          modulations
          )
        
        *
        2
        ⁢
        
          (
          PDP
          )
        
        *
        2
        ⁢
        
          
            (
            detections
            )
          
          ⁢
          
            
          
          [
          
            =
            32
          
          ]
        
      
    
    ⁢
    
      
    
    =
    
      
        
          112
          ⁢
          
            (
            optimum
            )
          
        
        +
        
          112
          ⁢
          
            (
            suboptimum
            )
          
        
      
      =
      
        224
        ⁢
        
          
        
        ⁢
        receiver
        ⁢
        
          
        
        ⁢
        designs
      
    
  
  ⁢
  
    
  




It will be appreciated by persons skilled in the art that the present invention may also include further additional modifications made to the device which does not affect the overall functioning of the device.
SEQUENCE LISTING
Not Applicable

## Claims

1. A broadcast receiver with improved response to long and short term fading which comprises:
a processor configured to receive a data signal which is assessed with respect to a decision metric derived upon utilising the probability density function of a Suzuki distribution in a closed form:





  
    
      p
      
        a
        l
      
    
    ⁡
    
      (
      
        a
        l
      
      )
    
  
  =
  
    
      
        (
        
          5
          2
        
        )
      
      
        1
        
          8
          ⁢
          
            σ
            2
          
        
      
    
    ⁢
    
      
        
          a
          1
        
        ⁢
        
          e
          
            -
            
              
                μ
                2
              
              
                2
                ⁢
                
                  σ
                  2
                
              
            
          
        
      
      
        σ
        ⁢
        
          
            2
            ⁢
            π
          
        
      
    
    ⁢
    
      Γ
      ⁡
      
        (
        
          1
          -
          
            1
            
              8
              ⁢
              
                σ
                2
              
            
          
          -
          
            μ
            
              2
              ⁢
              
                σ
                2
              
            
          
        
        )
      
    
    ⁢
    
      
        (
        
          
            -
            
              1
              
                8
                ⁢
                
                  σ
                  2
                
              
            
          
          -
          
            
              a
              l
              2
            
            2
          
        
        )
      
      
        
          -
          1
        
        +
        
          1
          
            8
            ⁢
            
              σ
              2
            
          
        
        +
        
          μ
          
            2
            ⁢
            
              σ
              2
            
          
        
      
    
  



where σ and μ denote the variance and mean value respectively of the Lognormal distribution, in order to take into account potential impairment of the said data signal.

2. A broadcast receiver according to claim 1 wherein the receiver is optimised in a plurality of conditions with respect to the geographical location to which the receiver is to be used.

3. A broadcast receiver according to claim 1 wherein the data signal impairment is with respect to the fading of the data signal channel.

4. A broadcast receiver according to claim 1 wherein when the amplitudes of the data signal are independent with first order probability density functions {pαl(al)}l=1Lp, and the phase and delay values are known, the decision metric derived from the Suzuki distribution is:


Λk=ln [p({rl(t)}l=1Lp|sk(t),{θl}l=1Lp,{τl}l=1Lp)].

5. A broadcast receiver according to claim 1 wherein the decision metric used when the phase value of the data signal is known but the amplitudes and delay values are not known is:



  
    
      Λ
      k
    
    =
    
      
        ln
        ⁡
        
          [
          
            p
            ⁡
            
              (
              
                
                  
                    
                      {
                      
                        
                          r
                          l
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      }
                    
                    
                      l
                      =
                      1
                    
                    Lp
                  
                  ❘
                  
                    
                      s
                      k
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                ,
                
                  
                    {
                    
                      θ
                      l
                    
                    }
                  
                  
                    l
                    =
                    1
                  
                  Lp
                
              
              )
            
          
          ]
        
      
      =
      
        ln
        (
        
          
            
              
                
                  K
                  ⁡
                  
                    (
                    
                      B
                      -
                      A
                    
                    )
                  
                
                ⁢
                
                  
                    ∏
                    
                      l
                      =
                      1
                    
                    
                      L
                      p
                    
                  
                  ⁢
                  
                    
                      
                        (
                        
                          5
                          2
                        
                        )
                      
                      
                        1
                        
                          8
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                    
                    ⁢
                    
                      
                        e
                        
                          -
                          
                            
                              μ
                              2
                            
                            
                              2
                              ⁢
                              
                                σ
                                2
                              
                            
                          
                        
                      
                      
                        2
                        ⁢
                        σ
                        ⁢
                        
                          
                            2
                            ⁢
                            π
                          
                        
                      
                    
                    ⁢
                    
                      Γ
                      ⁡
                      
                        (
                        
                          1
                          -
                          
                            1
                            
                              8
                              ⁢
                              
                                σ
                                2
                              
                            
                          
                          -
                          
                            μ
                            
                              2
                              ⁢
                              
                                σ
                                2
                              
                            
                          
                        
                        )
                      
                    
                  
                
              
            
          
          
            
              
                
                  
                    
                      [
                      
                        
                          N
                          l
                        
                        
                          E
                          k
                        
                      
                      ]
                    
                    
                      
                        -
                        1
                      
                      +
                      
                        
                          1
                          +
                          
                            4
                            ⁢
                            μ
                          
                        
                        
                          4
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                    
                  
                  ⁢
                  
                    Γ
                    ⁡
                    
                      (
                      
                        
                          
                            1
                            +
                            
                              4
                              ⁢
                              μ
                            
                          
                          
                            8
                            ⁢
                            
                              σ
                              2
                            
                          
                        
                        -
                        
                          1
                          2
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      F
                      1
                      
                      
                        1
                        
                    
                    ⁡
                    
                      (
                      
                        
                          
                            
                              1
                              +
                              
                                4
                                ⁢
                                μ
                              
                            
                            
                              8
                              ⁢
                              
                                σ
                                2
                              
                            
                          
                          -
                          
                            1
                            2
                          
                        
                        ,
                        
                          1
                          2
                        
                        ,
                        
                          
                            
                              e
                              
                                -
                                
                                  j2θ
                                  l
                                
                              
                            
                            ⁢
                            
                              
                                y
                                kl
                                2
                              
                              ⁡
                              
                                (
                                τ
                                )
                              
                            
                          
                          
                            4
                            ⁢
                            
                              E
                              k
                            
                            ⁢
                            
                              N
                              l
                            
                          
                        
                      
                      )
                    
                  
                
                +
              
            
          
          
            
              
                
                  
                     
                     
                  
                  ⁡
                  
                    [
                    
                      
                        
                          e
                          
                            -
                            
                              jθ
                              l
                            
                          
                        
                        ⁢
                        
                          
                            y
                            kl
                          
                          ⁡
                          
                            (
                            τ
                            )
                          
                        
                      
                      
                        N
                        l
                      
                    
                    ]
                  
                
                ⁢
                
                  
                    
                      N
                      l
                    
                    
                      E
                      k
                    
                  
                
                ⁢
                
                  Γ
                  ⁡
                  
                    (
                    
                      
                        1
                        +
                        
                          4
                          ⁢
                          μ
                        
                      
                      
                        8
                        ⁢
                        
                          σ
                          2
                        
                      
                    
                    )
                  
                
                ⁢
                
                  
                    F
                    1
                    
                    
                      1
                      
                  
                  ⁡
                  
                    (
                    
                      
                        
                          1
                          +
                          
                            4
                            ⁢
                            μ
                          
                        
                        
                          8
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                      ,
                      
                        3
                        2
                      
                      ,
                      
                        
                          
                            e
                            
                              -
                              
                                j2θ
                                l
                              
                            
                          
                          ⁢
                          
                            
                              y
                              kl
                              2
                            
                            ⁡
                            
                              (
                              τ
                              )
                            
                          
                        
                        
                          4
                          ⁢
                          
                            E
                            k
                          
                          ⁢
                          
                            N
                            l
                          
                        
                      
                    
                    )
                  
                
              
            
          
        
        )
      
    
  




  
    
      
    
    ⁢
    and
  




  
    
      Λ
      k
    
    =
    
      
        ln
        ⁡
        
          [
          
            p
            ⁡
            
              (
              
                
                  
                    
                      {
                      
                        
                          r
                          l
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      }
                    
                    
                      l
                      =
                      1
                    
                    Lp
                  
                  ❘
                  
                    
                      s
                      k
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                ,
                
                  
                    {
                    
                      θ
                      l
                    
                    }
                  
                  
                    l
                    =
                    1
                  
                  Lp
                
              
              )
            
          
          ]
        
      
      =
      
        

      
      ⁢
      
        
          ln
          (
          
            
              
                
                  
                    K
                    ⁡
                    
                      (
                      
                        B
                        -
                        A
                      
                      )
                    
                  
                  ⁢
                  
                    
                      ∏
                      
                        l
                        =
                        1
                      
                      
                        L
                        p
                      
                    
                    ⁢
                    
                      
                        
                          (
                          
                            5
                            2
                          
                          )
                        
                        
                          1
                          
                            8
                            ⁢
                            
                              σ
                              2
                            
                          
                        
                      
                      ⁢
                      
                        
                          e
                          
                            -
                            
                              
                                μ
                                2
                              
                              
                                2
                                ⁢
                                
                                  σ
                                  2
                                
                              
                            
                          
                        
                        
                          2
                          ⁢
                          σ
                          ⁢
                          
                            
                              2
                              ⁢
                              π
                            
                          
                        
                      
                      ⁢
                      
                        
                          
                            Γ
                            ⁡
                            
                              (
                              
                                1
                                -
                                
                                  1
                                  
                                    8
                                    ⁢
                                    
                                      σ
                                      2
                                    
                                  
                                
                                -
                                
                                  μ
                                  
                                    2
                                    ⁢
                                    
                                      σ
                                      2
                                    
                                  
                                
                              
                              )
                            
                          
                          ⁡
                          
                            [
                            
                              
                                N
                                l
                              
                              
                                E
                                k
                              
                            
                            ]
                          
                        
                        
                          
                            -
                            1
                          
                          +
                          
                            
                              1
                              +
                              
                                4
                                ⁢
                                μ
                              
                            
                            
                              4
                              ⁢
                              
                                σ
                                2
                              
                            
                          
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    Γ
                    ⁢
                    
                      (
                      
                        
                          
                            1
                            +
                            
                              4
                              ⁢
                              μ
                            
                          
                          
                            8
                            ⁢
                            
                              σ
                              2
                            
                          
                        
                        -
                        
                          1
                          2
                        
                      
                      )
                    
                    *
                    
                      2
                      
                        1
                        -
                        
                          1
                          
                            8
                            ⁢
                            
                              σ
                              2
                            
                          
                        
                        -
                        
                          μ
                          
                            2
                            ⁢
                            
                              σ
                              2
                            
                          
                        
                      
                    
                    ⁢
                    
                      
                        F
                        1
                        
                        
                          1
                          
                      
                      ⁡
                      
                        (
                        
                          
                            
                              
                                1
                                +
                                
                                  4
                                  ⁢
                                  μ
                                
                              
                              
                                8
                                ⁢
                                
                                  σ
                                  2
                                
                              
                            
                            -
                            
                              1
                              2
                            
                          
                          ,
                          
                            1
                            2
                          
                          ,
                          
                            
                              
                                e
                                
                                  -
                                  
                                    j2θ
                                    l
                                  
                                
                              
                              ⁢
                              
                                
                                  y
                                  kl
                                  2
                                
                                ⁡
                                
                                  (
                                  τ
                                  )
                                
                              
                            
                            
                              4
                              ⁢
                              
                                E
                                k
                              
                              ⁢
                              
                                N
                                l
                              
                            
                          
                        
                        )
                      
                    
                  
                  +
                
              
            
            
              
                
                  
                    [
                    
                      
                        
                          e
                          
                            -
                            
                              jθ
                              l
                            
                          
                        
                        ⁢
                        
                          
                            y
                            kl
                          
                          ⁡
                          
                            (
                            τ
                            )
                          
                        
                      
                      
                        N
                        l
                      
                    
                    ]
                  
                  ⁢
                  
                    
                      
                        N
                        l
                      
                      
                        E
                        k
                      
                    
                  
                  ⁢
                  
                    Γ
                    ⁡
                    
                      (
                      
                        
                          1
                          +
                          
                            4
                            ⁢
                            μ
                          
                        
                        
                          8
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      F
                      1
                      
                      
                        1
                        
                    
                    ⁡
                    
                      (
                      
                        
                          
                            1
                            +
                            
                              4
                              ⁢
                              μ
                            
                          
                          
                            8
                            ⁢
                            
                              σ
                              2
                            
                          
                        
                        ,
                        
                          3
                          2
                        
                        ,
                        
                          
                            
                              e
                              
                                -
                                
                                  j2θ
                                  l
                                
                              
                            
                            ⁢
                            
                              
                                y
                                kl
                                2
                              
                              ⁡
                              
                                (
                                τ
                                )
                              
                            
                          
                          
                            4
                            ⁢
                            
                              E
                              k
                            
                            ⁢
                            
                              N
                              l
                            
                          
                        
                      
                      )
                    
                  
                
              
            
          
          )
        
        .

6. A broadcast receiver according to claim 1 wherein the decision metric used in the case of known delay and amplitude values of the data signal but the phase value is unknown is:




  
    
      
        
          Λ
          k
        
        =
        
          ln
          ⁡
          
            [
            
              p
              ⁡
              
                (
                
                  
                    
                      
                        {
                        
                          
                            r
                            l
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        }
                      
                      
                        l
                        =
                        1
                      
                      Lp
                    
                    ❘
                    
                      
                        s
                        k
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  ,
                  
                    
                      {
                      
                        α
                        l
                      
                      }
                    
                    
                      l
                      =
                      1
                    
                    Lp
                  
                  ,
                  
                    
                      {
                      
                        τ
                        l
                      
                      }
                    
                    
                      l
                      =
                      1
                    
                    Lp
                  
                
                )
              
            
            ]
          
        
      
    
  
  
    
      
        =
        
          K
          ⁢
          
            
              ∑
              
                l
                =
                1
              
              
                L
                p
              
            
            ⁢
            
              
                ln
                ⁡
                
                  [
                  
                    
                      e
                      
                        -
                        
                          
                            
                              a
                              l
                              2
                            
                            ⁢
                            
                              E
                              k
                            
                          
                          
                            N
                            l
                          
                        
                      
                    
                    ⁢
                    
                      
                        I
                        o
                      
                      ⁡
                      
                        (
                        
                          
                            
                              a
                              l
                            
                            
                              N
                              l
                            
                          
                          ⁢
                          
                            
                            
                              
                                y
                                kl
                              
                              ⁡
                              
                                (
                                
                                  τ
                                  l
                                
                                )
                              
                            
                            
                          
                        
                        )
                      
                    
                  
                  ]
                
              
              .

7. A broadcast receiver according to claim 1 wherein the decision metric used in the case of a known delay value in the data signal but unknown amplitude and phase values is:


Λk=ln [p({rl(t)}l=1Lp|sk(t),{τl}l=1Lp)].

8. A broadcast receiver according to claim 1 wherein the decision metric used in the case when the delay, amplitude and phase values are unknown is:



  
    
      Λ
      k
    
    =
    
      
        ln
        ⁡
        
          [
          
            p
            ⁡
            
              (
              
                
                  
                    {
                    
                      
                        r
                        l
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    }
                  
                  
                    l
                    =
                    1
                  
                  Lp
                
                ❘
                
                  
                    s
                    k
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
              )
            
          
          ]
        
      
      =
      
        
          ln
          [
          
            
              K
              ⁡
              
                (
                
                  B
                  -
                  A
                
                )
              
            
            ⁢
            
              
                ∏
                
                  l
                  =
                  1
                
                
                  L
                  p
                
              
              ⁢
              
                
                  
                    (
                    
                      5
                      2
                    
                    )
                  
                  
                    1
                    
                      8
                      ⁢
                      
                        σ
                        2
                      
                    
                  
                
                ⁢
                
                  
                    e
                    
                      -
                      
                        
                          μ
                          2
                        
                        
                          2
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                    
                  
                  
                    σ
                    ⁢
                    
                      
                        2
                        ⁢
                        n
                      
                    
                  
                
                ⁢
                
                  Γ
                  ⁡
                  
                    (
                    
                      1
                      -
                      
                        1
                        
                          8
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                      -
                      
                        μ
                        
                          2
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                    
                    )
                  
                
              
            
          
          ]
        
        ⁢
        
          
            
              1
              2
            
            ⁡
            
              [
              
                
                  E
                  k
                
                
                  N
                  l
                
              
              ]
            
          
          
            -
            
              
                1
                +
                
                  4
                  ⁢
                  μ
                
              
              
                8
                ⁢
                
                  σ
                  2
                
              
            
          
        
        ⁢
        
          Γ
          ⁡
          
            (
            
              
                1
                +
                
                  4
                  ⁢
                  μ
                
              
              
                8
                ⁢
                
                  σ
                  2
                
              
            
            )
          
        
      
    
  




  
    
      
    
    ⁢
    and
  




  
    
      
    
    ⁢
    
      
        ln
        [
        
          
            
              
                K
                ⁢
                
                  (
                  
                    B
                    -
                    A
                  
                  )
                
                ⁢
                
                  
                    ∏
                    
                      l
                      =
                      1
                    
                    
                      L
                      p
                    
                  
                  ⁢
                  
                    
                      
                        (
                        
                          5
                          2
                        
                        )
                      
                      
                        1
                        
                          8
                          ⁢
                          
                            σ
                            2
                          
                        
                      
                    
                    ⁢
                    
                      
                        e
                        
                          -
                          
                            
                              μ
                              2
                            
                            
                              2
                              ⁢
                              
                                σ
                                2
                              
                            
                          
                        
                      
                      
                        4
                        ⁢
                        σπ
                        ⁢
                        
                          2
                        
                      
                    
                    ⁢
                    
                      Γ
                      ⁡
                      
                        (
                        
                          1
                          -
                          
                            1
                            
                              8
                              ⁢
                              
                                σ
                                2
                              
                            
                          
                          -
                          
                            μ
                            
                              2
                              ⁢
                              
                                σ
                                2
                              
                            
                          
                        
                        )
                      
                    
                  
                
              
            
          
          
            
              
                
                  
                    [
                    
                      
                        E
                        k
                      
                      
                        N
                        l
                      
                    
                    ]
                  
                  
                    
                      1
                      4
                    
                    -
                    
                      
                        1
                        +
                        
                          4
                          ⁢
                          μ
                        
                      
                      
                        8
                        ⁢
                        
                          σ
                          2
                        
                      
                    
                  
                
                ⁢
                
                  2
                  
                    
                      1
                      2
                    
                    -
                    
                      1
                      
                        8
                        ⁢
                        
                          σ
                          2
                        
                      
                    
                    -
                    
                      μ
                      
                        2
                        ⁢
                        
                          σ
                          2
                        
                      
                    
                  
                
                ⁢
                
                  
                    
                      
                      
                        
                          y
                          kl
                        
                        ⁡
                        
                          (
                          
                            τ
                            l
                          
                          )
                        
                      
                      
                    
                    
                      N
                      l
                    
                  
                
              
            
          
          
            
              
                (
                
                  
                    
                      
                        
                          
                            Γ
                            ⁡
                            
                              (
                              
                                
                                  
                                    1
                                    +
                                    
                                      4
                                      ⁢
                                      μ
                                    
                                  
                                  
                                    8
                                    ⁢
                                    
                                      σ
                                      2
                                    
                                  
                                
                                -
                                
                                  1
                                  4
                                
                              
                              )
                            
                          
                          ⁢
                          
                            
                              F
                              1
                              
                              
                                1
                                
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    
                                      1
                                      +
                                      
                                        4
                                        ⁢
                                        μ
                                      
                                    
                                    
                                      8
                                      ⁢
                                      
                                        σ
                                        2
                                      
                                    
                                  
                                  -
                                  
                                    1
                                    4
                                  
                                
                                ,
                                
                                  1
                                  2
                                
                                ,
                                
                                  
                                    
                                      y
                                      kl
                                      2
                                    
                                    ⁡
                                    
                                      (
                                      τ
                                      )
                                    
                                  
                                  
                                    4
                                    ⁢
                                    
                                      E
                                      k
                                    
                                    ⁢
                                    
                                      N
                                      l
                                    
                                  
                                
                              
                              )
                            
                          
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          
                             
                             
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  y
                                  kl
                                
                                ⁡
                                
                                  (
                                  τ
                                  )
                                
                              
                              
                                
                                  
                                    E
                                    k
                                  
                                  ⁢
                                  
                                    N
                                    l
                                  
                                
                              
                            
                            ]
                          
                        
                        ⁢
                        
                          Γ
                          ⁡
                          
                            (
                            
                              
                                
                                  1
                                  +
                                  
                                    4
                                    ⁢
                                    μ
                                  
                                
                                
                                  8
                                  ⁢
                                  
                                    σ
                                    2
                                  
                                
                              
                              +
                              
                                1
                                4
                              
                            
                            )
                          
                        
                        ⁢
                        
                          
                            F
                            1
                            
                            
                              1
                              
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  
                                    1
                                    +
                                    
                                      4
                                      ⁢
                                      μ
                                    
                                  
                                  
                                    8
                                    ⁢
                                    
                                      σ
                                      2
                                    
                                  
                                
                                +
                                
                                  1
                                  4
                                
                              
                              ,
                              
                                3
                                2
                              
                              ,
                              
                                
                                  
                                    y
                                    kl
                                    2
                                  
                                  ⁡
                                  
                                    (
                                    τ
                                    )
                                  
                                
                                
                                  4
                                  ⁢
                                  
                                    E
                                    k
                                  
                                  ⁢
                                  
                                    N
                                    l
                                  
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                
                )
              
            
          
        
        ]
      
      .

9. A broadcast data receiver according to claim 1 wherein the configuration of the receiver includes details relating to coherent detection as well as non-coherent detection.

10. A broadcast receiver according to claim 1 wherein the configuration of the receiver includes details of possible modulation schemes to be used.

11. A broadcast receiver according to claim 1 wherein the receiver is configured to receive data signals which are provided using shaping pulses.

12. A method of configuring receiving and/or processing apparatus for at least one data signal received from a remote transmitting location, said method comprising the steps of:
identifying a data signal to be received; and
configuring a receiving and/or processing apparatus based on said identified signal with reference to a decision metric derived upon utilising the probability density function of a Suzuki distribution in a closed form:





  
    
      p
      
        a
        l
      
    
    ⁡
    
      (
      
        a
        l
      
      )
    
  
  =
  
    
      
        (
        
          5
          2
        
        )
      
      
        1
        
          8
          ⁢
          
            σ
            2
          
        
      
    
    ⁢
    
      
        
          a
          1
        
        ⁢
        
          e
          
            -
            
              
                μ
                2
              
              
                2
                ⁢
                
                  σ
                  2
                
              
            
          
        
      
      
        σ
        ⁢
        
          
            2
            ⁢
            π
          
        
      
    
    ⁢
    
      Γ
      ⁡
      
        (
        
          1
          -
          
            1
            
              8
              ⁢
              
                σ
                2
              
            
          
          -
          
            μ
            
              2
              ⁢
              
                σ
                2
              
            
          
        
        )
      
    
    ⁢
    
      
        (
        
          
            -
            
              1
              
                8
                ⁢
                
                  σ
                  2
                
              
            
          
          -
          
            
              a
              l
              2
            
            2
          
        
        )
      
      
        
          -
          1
        
        +
        
          1
          
            8
            ⁢
            
              σ
              2
            
          
        
        +
        
          μ
          
            2
            ⁢
            
              σ
              2
            
          
        
      
    
  



where σ and μ denote the variance and mean value respectively of the Lognormal distribution, in order to take into account potential impairment of the said data signal.

13. A method according to claim 12 wherein the method includes the step of identifying the geographical location at which the said apparatus is to be used.